# We can display an object on a monitor screen in three different computer-model forms: Wireframe model Surface Model Solid model

Save this PDF as:

Size: px
Start display at page:

Download "We can display an object on a monitor screen in three different computer-model forms: Wireframe model Surface Model Solid model"

## Transcription

1 CHAPTER 4 CURVES 4.1 Introduction In order to understand the significance of curves, we should look into the types of model representations that are used in geometric modeling. Curves play a very significant role in CAD modeling, especially, for generating a wireframe model, which is the simplest form for representing a model. We can display an object on a monitor screen in three different computer-model forms: Wireframe model Surface Model Solid model Wireframe model: A wireframe model consist of points and curves only, and looks as if its made up with a bunch of wires. This is the simplest CAD model of an object. Advantages of this type of model include ease of creation and low level hardware and software requirements. Additionally, the data storage requirement is low. The main disadvantage of a wireframe model is that it can be very confusing to visualize. For example, a blind hole in a box may look like a solid cylinder, as shown in the figure. A wireframe model Model of a Solid object with a blind hole ME 165 Lecture Notes by R. B. Agarwal Computer Aided Design in Mechanical Engineering 4-1

2 In spite of its ambiguity, a wireframe model is still the most preferred form, because it can be created quickly and easily to verify a concept of an object. The wireframe model creation is somewhat similar to drawing a sketch by hand to communicate or conceptualize an object. As stated earlier, a wireframe model is created using points and curves only. Surface Model: sweeping a curve around or along an axis can create a surface model. The figures below show two instances of generating a surface model. Generating a cylinder by sweeping a circle in the direction of an axis generating a donut by sweeping a circle around an axis The appearance or resolution of a surface model depends on the number of sweeping instances we select. For a realistic looking model, we need to select a large number of instances, requiring a large computer memory, or, opt for a not-so realistic model by selecting a small number of instances, and save memory. In some commercial CAD packages we have the option of selecting the resolution of a model, other packages have a fixed value for resolution that cannot be changed by users. Surface models are useful for representing surfaces such as a soft-drink bottle, automobile fender, aircraft wing, and in general, any complicated curved surface. One of the limitations of a surface model is that there is no geometric definition of points that lie inside or outside the surface. Solid Model: Representation of an object by a solid model is relatively a new concept. There were only a couple of solids modeling CAD programs available in late 1980s, and they required mainframe computers to run on. However, in 1990s, due to the low cost and high speed, PCs have become the most popular solid modeling software platform, prompting almost all the CAD vendors to introduce their 3-D solid modeling software that will run on a PC. Solid models represent objects in a very realistic and unambiguous form; however, they require a large amount of storage memory and high-end computer hardware. A solid model can be shaded and rendered in desired colors to give it a more realist appearance. ME 165 Lecture Notes by R. B. Agarwal Computer Aided Design in Mechanical Engineering 4-2

3 4.2 Role of Curves in Geometric Modeling Curves are used to draw a wireframe model, which consists of points and curves; the curves are utilized to generate surfaces by performing parametric transformations on them. A curve can be as simple as a line or as complex as a B-spline. In general, curves can be classified as follows: Analytical Curves: This type of curve can be represented by a simple mathematical equation, such as, a circle or an ellipse. They have a fixed form and cannot be modified to achieve a shape that violates the mathematical equations. Interpolated curves: An interpolated curve is drawn by interpolating the given data points and has a fixed form, dictated by the given data points. These curves have some limited flexibility in shape creation, dictated by the data points. Approximated Curves: These curves provide the most flexibility in drawing curves of very complex shapes. The model of a curved automobile fender can be easily created with the help of approximated curves and surfaces. In general, sweeping a curve along or around an axis creates a surface, and the generated surface will be of the same type as the generating curve, e.g., a fixed form curve will generate a fixed form surface. As stated earlier, curves are used to generate surfaces. To facilitate the computer-language algorithm, curves are represented by parametric equations. Non-parametric equations are used only to locate a point of intersection on the curve, and not for generating them. Let us briefly discus the parametric and non-parametric form of a curve. ME 165 Lecture Notes by R. B. Agarwal Computer Aided Design in Mechanical Engineering 4-3

4 4.3 Parametric and Non-parametric Equations of a Curve The mathematical representation of a curve can be classified as either parametric or nonparametric (natural). A non-parametric equation has the form, y = c 1 + c 2 x + c 3 x 2 + c 4 x 3 Explicit non-parametric equation This is an example of an explicit non-parametric curve form. In this equation, there is a unique single value of the dependent variable for each value of the independent variable. The implicit non-parametric form of an equation is, (x x c ) 2 + (y y c ) 2 = r 2 Implicit non-parametric equation In this equation, no distinction is made between the dependent and the independent variables. Parametric Equations: Parametric equations describe the dependent and independent variables in terms of a parameter. The equation can be converted to a non-parametric form, by eliminating the dependent and independent variables from the equation. Parametric equations allow great versatility in constructing space curves that are multi-valued and easily manipulated. Parametric curves can be defined in a constrained period (0 t 1); since curves are usually bounded in computer graphics, this characteristic is of considerable importance. Therefore, parametric form is the most common form of curve representation in geometric modeling. Examples of parametric and non-parametric equations follow. Non-Parametric Parametric Circle: x 2 + y 2 = r 2 x = r cosθ, y = r sinθ Where, θ is the parameter. CAD programs prefer a parametric equation for generating a curve. Parametric equations are converted into matrix equations to facilitate a computer solution, and then varying a parameter from 0 to 1 creates the points or curves. In this course, we will use the following parameters, with the range indicated, 0 t 1 0 s 1 0 θ 2πs 0 ϕ 2πs ME 165 Lecture Notes by R. B. Agarwal Computer Aided Design in Mechanical Engineering 4-4

5 4.4 Fixed-Form or Analytical Curves Equation of a Straight Line: The simplest fixed-form curve is a straight line. Parametric equation of a straight line is given as, P(t) = A + (B-A) t (4.1) The parametric equation of line AB can be derived as, x = x 1 + (x 2 - x 1 ) t y = y 1 + (y 2 - y 1 ) t. P (x, y) B (x 2, y 2 ) where, 0 t 1 A (x 1, y 1 ) The point P on the line is sweeped from A to B, as the value of t is varied from 0 to Conic Sections or Conic Curves A conic curve is generated when a plane intersects a cone, as shown. B P Q A A ME 165 Lecture Notes by R. B. Agarwal Computer Aided Design in Mechanical Engineering 4-5

6 The intersection of the plane PQ and the cone is a circle, where as, the intersection created by the plane AB is an ellipse. Other curves that can be created are parabola and hyperbola. Conic curves are used to create simple wireframe models of objects, which have edges that can be represented by these analytical curves. The fixed-form or analytical curves do not have inflection points, i.e., curves have slopes that are either positive or negative and do not change their sign (positive slope will remain positive and negative slope will remain negative). All conic curves can be represented by a quadratic equation, for example, circular and elliptical curves have quadratic polynomial equations Circular Curve The non-parametric equation of a circle is, (x x c ) 2 + (y y c ) 2 = r 2 (4.2) Where, x c, and y c are coordinates of the center, and r is radius of the circle. If we were to use this form of the equation for plotting a circle or a circular curve, we will first calculate several values of x and y along the circumference of the circle, and then plot them. The curve thus generated will be of a poor quality, unless we plot a very large number of data points, which will result in a significant demand for storage of these data points. Therefore, as stated earlier, in CAD programs, we use a parametric equation, which avoids the need for storage of the data points, and provides a smooth curve. The parametric equation of the above circle can be written as, x i = x c + r cosθ y i = y c + r sinθ (4.3) This equation is converted into a matrix form so that a computer can solve it. We will now convert this equation into a matrix form. Let us assume that the plot starts at the point (x i, y i ), and the center lies at the origin. We increment θ to (θ + θ), giving us the new point on the circle (x i+1, y i+1 ), or x i+1 = r cos(θ + θ) y i+1 = r sin(θ + θ) ME 165 Lecture Notes by R. B. Agarwal Computer Aided Design in Mechanical Engineering 4-6

7 (x i+1, y i+1 ) θ (x i, y i ) θ Expanding it by the use of trigonometric identities, we get: x i+1 = r cosθ cos θ - r sinθ sin θ y i+1 = r sinθ cos θ + r cosθ sin θ Substituting the values: x i = r cosθ, and y i = r sinθ, we get x i+1 = x i cos θ - y i sin θ y i+1 = y i cos θ + x i sin θ In matrix form, these equations can be written as, cos θ sin θ 0 0 [x i+1 y i+1 0 1] = [x i y i 0 1] - sin θ cos θ 0 0 (4.4) Equation (4.4) is valid for a circle that has center at the origin. To find the equation of a circle that has center located at an arbitrary point (x c, y c ), we can use the translation transformation. Note that the equation (4.4) can be interpreted as rotational transformation of points x i and y i about the origin. Now, instead of rotation about the origin, we wish to rotate the point about the fixed point (x c, y c ). This can be accomplished by the three-step approach, discussed in chapter 2, i.e., first translate the fixed point to the origin, rotate the object, and finally translate it so that the fixed point is restored to its original position. Using this procedure we will get: ME 165 Lecture Notes by R. B. Agarwal Computer Aided Design in Mechanical Engineering 4-7

8 [x i+1 y i+1 0 1] = cos θ sin θ 0 0 [x i y i 0 1] sin θ cos θ x c -y c (4.5) x c y c 0 1 Simplifying the equation we get, x i+1 = x c + (x i x c ) cos θ - (y i y c ) sin θ y i+1 = y c + (x i x c ) sin θ + (y i y c ) cos θ (4.6) Even though, equations (4.6) can be used as an iterative formula to plot a circle or a circular curve, using the EXCEL or MATLAB, or any other plot routines, the matrix equation (4.5) is the preferred form for a CAD program. The original CAD programs used iterative formulas to generate curves. The BASIC and FORTRAN languages were used to write the CAD codes. ME 165 Lecture Notes by R. B. Agarwal Computer Aided Design in Mechanical Engineering 4-8

9 4.4.4 Ellipse Following the procedure outlined in the previous section, we can derive the parametric equations of an ellipse. Parametric equation of an ellipse is given by the equation x i = a cosθ y i = b sinθ 2b For a point on the ellipse, in general, the equation is x i+1 = x i cos θ (a/b) y i sin θ y i+1 = y i cos θ (b/a) x i sin θ (4.7) 2a For a more general case, when the axes of the ellipse are not parallel to the coordinate axes, and the center of the ellipse is at a distance x c, y c from the origin, the equation of the ellipse is given below. Let α be the angle that the major axis makes with the horizontal (x-axis), as shown. The equation of the ellipse can be derived as x i = x c + x i cosα y i sinα y i = y c + x i sinα + y i cosα (4.8) Where, x and y are the coordinate values of a Point on the ellipse, in term of the rotated axes x and y. Equations (4.8) can be used to write either as an iterative formula or as a matrix equation for creating an elliptical curve. ME 165 Lecture Notes by R. B. Agarwal Computer Aided Design in Mechanical Engineering 4-9

10 4.5 Interpolated Curves Interpolation method can be applied to draw curves that pass through a set of the given data points. The resulting curve can be a straight line, quadratic, cubic, or higher order curve. We are quite familiar, and have used, the linear interpolation of a straight line, given by the formula f(x) = f(x i ) + [f(x i+1 ) f(x i )] [(x-x i ) / (x i+1 x i )] (4.9) Now, we will discuss the higher order curves, which are represented by higher order polynomials. Lagrange polynomial is a popular polynomial function used for interpolation of high order polynomials Lagrange Polynomial When a sequence of planar points (x 0, y 0 ), (x 1, y 1 ), (x 2, y 2 ),.(x n, y n ) is given, the n th degree of interpolated polynomial can be calculated by the Lagrange Polynomial equation, f n (x) = Σ y i L i,n (x) (4.10) where, L i,n (x) = [(x x 0 ). (x x i-1 ) (x x i+1 ). (x x n )] / [(x i x 0 ). (x i x i-1 ) (x i x i+1 ). (x i x n )] To understand the above expression better, note that The term (x x i ) is skipped in the numerator, and The denominator starts with the term (x i x 0 ) and skips the term (x i x i ), which will make the expression equal to infinity. Example: Using the Lagrange polynomial, find the expression of the curve containing the points, P 0 (1, 1), P 1 (2, 2), P 2 (3, 1) Solution: Here, n = 2 and x 0 =1, y 0 = 1, x 1 = 2, y 1 = 2, etc. The polynomial is of a second degree. Expanding the Lagrange equation, we get, f 2 (x) = y 0 [(x - x 1 ) (x - x 2 )] / [(x 0 x 1 ) (x 0 x 2 )] + y 1 [(x x 0 ) (x - x 2 )] / [(x 1 x 0 ) (x 1 x 2 )] + y 2 [(x x 0 ) (x x 1 )] / [(x 2 x 0 ) (x 2 x 1 )] ME 165 Lecture Notes by R. B. Agarwal Computer Aided Design in Mechanical Engineering 4-10

11 = (1) [(x 2) (x 3)] / [(1 2) (1 3)] + (2) [(x-1) (x 3)] / [(2 1) (2 3)] + (1) [(x 1) (x - 2)] / [(3 1) (3-2)] = ½ (x 2 5x + 6) 2 (x 2 4 x + 3) + ½ (x 2 3 x + 2) or f 2 (x) = - x x 2 This is the explicit non-parametric equation of a circle; the given points lie on the circumference Parametric Cubic Curve or Cubic Spline Synthetic Curves The analytical and interpolated curves, discussed in the previous section (4.4) and (4.5) are insufficient to meet the requirements of mechanical parts that have complex curved shapes, such as, propeller blades, aircraft fuselage, automobile body, etc. These components contain nonanalytical, synthetic curves. Design of curved boundaries and surfaces require curve representations that can be manipulated by changing data points, which will create bends and sharp turns in the shape of the curve. The curves are called synthetic curves, and the data points are called vertices or control points. If the curve passes through all the data points, it is called an interpolant (interpolated). Smoothness of the curve is the most important requirement of a synthetic curve. Various continuity requirements at the data points can be specified to impose various degrees of smoothness of the curve. A complex curve may consist of several curve segments joined together. Smoothness of the resulting curve is assured by imposing one of the continuity requirements. A zero order continuity (C 0 ) assures a continuous curve, first order continuity (C 1 ) assures a continuous slope, and a second order continuity (C 2 ) assures a continuous curvature, as shown below. C 0 Continuity The curve is C 1 Continuity- Slope Continuity C 2 Continuity - Curvature Continuous everywhere at the common point continuity at the common point A cubic polynomial is the lowest degree polynomial that can guarantee a C 2 curve. Higher order polynomials are not used in CAD, because they tend to oscillate about the control points and require large data storage. Major CAD/CAM systems provide three types of synthetic curves: Hermite Cubic Spline, Bezier Curves, and B-Spline Curves. ME 165 Lecture Notes by R. B. Agarwal Computer Aided Design in Mechanical Engineering 4-11

12 Cubic Spline curves pass through all the data points and therefore they can be called as interpolated curves. Bezier and B-Spline curves do not pass through all the data points, instead, they pass through the vicinity of these data points. Both the cubic spline and Bezier curve have first-order continuity, where as, B-Spline curves have a second-order continuity Hermite Cubic Spline Hermite cubic curve is also known as parametric cubic curve, and cubic spline. This curve is used to interpolate given data points that result in a synthetic curve, but not a free form, unlike the Bezier and B-spline curves. The most commonly used cubic spline is a three-dimensional planar curve (not twisted). The curve is defined by two data points that lie at the beginning and at the end of the curve, along with the slopes at these points. It is represented by a cubic polynomial. When two end points and their slopes define a curve, the curve is called a Hermite cubic curve. Several cubic splines can be joined together by imposing the slope continuity at the common points. In design applications, cubic splines are not as popular as the Bezier and B-spline curves. There are two reasons for this: The curve cannot be modified locally, i.e., when a data point is moved, the entire curve is affected, resulting in a global control, as shown in the figure. The order of the curve is always constant (cubic), regardless of the number of data points. Increase in the number of data points increases shape flexibility, However, this requires more data points, creating more splines, that are joined together (only two data points and slopes are utilized for each spline). Effect of Moving the Data Point Effect of Change in slope Equation of a Cubic Spline A cubic spline is a third-degree polynomial, defined as P(t) = Σ a i t i (4.11) where, 0 t 1, and P(t) is a point on the curve. Expanding the above equation, we get ME 165 Lecture Notes by R. B. Agarwal Computer Aided Design in Mechanical Engineering 4-12

13 P(t) = a 3 t 3 + a 2 t 2 + a 1 t + a 0 (4.12) If (x,y,z) are the coordinates of point P, the equation (4.12) can be written as, x(t) = a 3x t 3 + a 2x t 2 + a 1x t + a 0x y(t) = a 3y t 3 + a 2y t 2 + a 1y t + a 0y (4.13) z(t) = a 3z t 3 + a 2z t 2 + a 1z t + a 0z There are 12 unknown coefficients, a ij, known as the algebraic coefficients. These coefficients can be evaluated by applying the boundary conditions at the end points. From the coordinates of the end points of each segment, six of the twelve needed equations are obtained. The other six equations are found by using the tangent vectors at the two ends of each segment. Substituting the boundary conditions at t = 0, and t = 1, we get, P(0) = a 0, and P(1) = a 3 + a 2 + a 1 + a 0 (a) (b) To find the tangent vectors, we differentiate equation (4.12), and get, P (t) = 3 a 3 t a 2 t + a 1 Applying the boundary conditions at t = 0 and t = 1, we get, P (0) = a 1 P (1) = 3 a a 2 + a 1 (c) (d) Solving for the coefficients in terms of the P(t) and P (t) values in equations (a) through (d), we get, a 0 = P(0) a 1 = P (0) a 2 = -3 P(0) +3 P(1) 2 P (0) P (1) (4.14) a 3 = 2 P(0) 2 P(1) + P (0) + P (1) ME 165 Lecture Notes by R. B. Agarwal Computer Aided Design in Mechanical Engineering 4-13

14 The equation P(t) = a 3 t 3 + a 2 t 2 + a 1 t + a 0 can be written, with coefficients a ij replaced by the P(t) and P (t) values in equations (4.14), resulting, P(t) = [2 P(0) 2 P(1) + P (0) + P (1)] t 3 + [-3 P(0) + 3 P(1) 2 P (0) P (1)] t 2 + P (0) t + P(0) Or, rearranging the terms, we get, P(t) = [(2 t 3 3 t 2 + 1)] P(0) + [(-2 t t 2 )] P(1) + [(t 3 2 t 2 + t)] P (0) + [(t 3 t 2 )] P (1) In matrix form the equation can be written as, P(0) P(t) = [t 3 t 2 t 1] P(1) P (0) (4.15) P (1) The equation in short form can be written as: P(t) = [t] [M] H [G] Where, the terms [t], [M] H, and [G] correspond to the terms on the right hand side of the equation (4.15). [M] H is called Hermite matrix of a cubic spline, and represents the constant matrix. The term [G] is called geometric coefficient matrix. Let us consider an example to understand how the equation (4.15) works. Example 5: A parametric cubic curve passes through the points (0,0), (2,4), (4,3), (5, -2) which are parametrized at t = 0, ¼, ¾, and 1, respectively. Determine the geometric coefficient matrix and the slope of the curve when t = 0.5. Solution: The points on the curve are (0,0) at t = 0 (2,4) at t = ¼ ME 165 Lecture Notes by R. B. Agarwal Computer Aided Design in Mechanical Engineering 4-14

15 (4,3) at t = ¾ (5,-2) at t = 1 Substituting in equation (4.15), we get, P(0) P(1) 4 3 = P (0) P (1) Solving, we get, P(0) 0 0 P(1) 5-2 P (0) = P (1) The slope at t = 0.5 is found by taking the first derivative of the equation (4.15), as follows, Therefore, P (t) = [3t 3 2t 1 0] P (0.5) = [ ], or Slope = x/ y = -2.0/3.67 = Note: coinciding the end points, and imposing equal values of the slopes, as shown, can create closed shape of a cubic spline. ME 165 Lecture Notes by R. B. Agarwal Computer Aided Design in Mechanical Engineering 4-15

16 4.6 Approximated Synthetic Curves In the previous sections, we have studied the analytical and interpolated curves, now we will focus on the approximated curves. Bezier and B-spline curves represent the approximated curves, these curves are synthetic, and can be joined together to form a very smooth curve. For data fitting, interpolated curves work best, where as, for free form geometry, interpolation cannot be used, and approximation becomes necessary. In many engineering applications, smoothness of a curve is preferred over the quality of interpolation. These curves are flexible, local changes in the shape do not affect the entire shape of the curve. Let us study the Bezier curve first, followed by the B-spline Bezier Curves Equation of the Bezier curve provides an approximate polynomial that passes near the given control points and through the first and last points. In 1960s, the French engineer P. Bezier, while working for the Renault automobile manufacturer, developed a system of curves that combine the features of both interpolating and approximating polynomials. In this curve, the control points influence the path of the curve and the first two and last two control points define lines which are tangent to the beginning and the end of the curve. Several curves can be combined and blended together. In engineering, only the quadratic, cubic and quartic curves are frequently used Bezier s Polynomial Equation The curve is defined by the equation P(t) = V i B i,n (t) where, 0 t 1 and i = 0, 1, 2,, n (4.16) Here, V i represents the n+1 control points, and B i,n (t) is the blending function for the Bezier representation and is given as n B i,n (t) = i ( t i ) (1-t) n-i (4.17) Where n is the degree of the polynomial and n n! i = i = 0, 1, 2,.,n i! (n-i)! These blending functions satisfy the following equations ME 165 Lecture Notes by R. B. Agarwal Computer Aided Design in Mechanical Engineering 4-16

17 B i,n (t) > 0 for all i B i,n (t) = 1 (4.18) The equations (4.18) force the curve to lie entirely within the convex figure (or envelop) set by the extreme points of the polygon formed by the control points. The envelope represents the figure created by stretching a rubber band around all the control points. The figure below shows that the first two points and the last two points form lines that are tangent to the curve. Also, as we move point v 1, the curve changes shape, such that the tangent lines always remain tangent to the curve. v 1 v 2 v 1 v 0 v 3 Relationship between end-points and curve slope Bezier s blending function produces an nth degree polynomial for n+1 control points and forces the Bezier curve to interpolate the first and last control points. The intermediate control points pull the curve toward them, and can be used to adjust the curve to the desired shape. ME 165 Lecture Notes by R. B. Agarwal Computer Aided Design in Mechanical Engineering 4-17

18 4.6.3 Third Order Bezier Polynomial We will simplify the Bezier s equation for n = 3 (a cubic curve). The procedure developed here can be extended to the other values of n. For n = 3, we will have four control points, namely, V 0, V 1, V 2, V 3. i will vary from 0 to 3. The Bezier s equation, P(t) = V i B i,3 (t) can be expanded to give, (4.19) P(t) = V 0 B 0,3 + V 1 B 1,3 + V 2 B 2,3 + V 3 B 3,3 and 3! B 0,3 = t 0 (1-t) 3 = (1-t) 3 0! 3! 3! B 1,3 = t 1 (1-t) 2 = 3t (1-t) 2 1! 2! 3! B 2,3 = ! 1! t 2 (1-t) 1 = 3t 2 (1-t) 3! B 3,3 = t 3 (1-t) 0 = t 3 3! 0! By substituting the above values in the equation (4.19) we get P(t) = (1-t) 3 V 0 +3t (1-t) 2 V 1 + 3t 2 (1-t) V 2 + t 3 V 3 In matrix form this equation is written as V 0 P(t) = [t 3 t 2 t 1] V 1 (4.20) V V 3 ME 165 Lecture Notes by R. B. Agarwal Computer Aided Design in Mechanical Engineering 4-18

19 4.6.4 Blending Two or More Bezier Curves Two or more Bezier curves can be blended to provide a desired curve of a complex nature. When joining curves, slope continuity is maintained by having three collinear points, the middle one being common to the adjoining curves, as shown. Point V 3 is the middle point of the common points V 2,, V 3,, and V 4 of curves A and B. NOTE: Using the Bezier curves, we can create closed curves by making the first and last points of the control points coincide. Example: A cubic Bezier curve is described by the four control points: (0,0), (2,1), (5,2), (6,1). Find the tangent to the curve at t = 0.5. Solution: We will use the Bezier cubic polynomial, given in equation (4.20), which is, V 0 P(t) = [t 3 t 2 t 1] V V V 3 ME 165 Lecture Notes by R. B. Agarwal Computer Aided Design in Mechanical Engineering 4-19

20 where, V 0 = (0,0) V 1 = (2,1) V 2 = (5,2) V 3 = (6,1) The tangent is given by the derivative of the general equation above, V 0 P (t) = [3t 2 2t 1 0] V 1 (4.21) V V 3 At t = 0.5, we get, V 0 P (t) = [3(.5) 2 2(.5) 1 0] V V V 3 = [ ] ME 165 Lecture Notes by R. B. Agarwal Computer Aided Design in Mechanical Engineering 4-20

21 4.6.5 B-Spline Curve B-spline curves use a blending function, which generates a smooth, single parametric polynomial curve through any number of points. To generate a Bezier curve of the same quality of smoothness, we will have to use several pieces of Bezier curves. Unlike the Bezier curve, the degree of the polynomial can be selected independently of the number of control points. The degree of the blending function controls the degree of the resulting B-spline curve. The curve has good local control, i.e., if one vertex is moved, only some curve segments are affected, and the rest of the curve remains unchanged. The mathematical derivation of the B-spline curve is complex and beyond the scope of this course. The equation is of the form: P(t) = Σ N i,k (t) V i (4.22) Where, P(t) is a point on the curve. i indicates the position of control point i k is order of curve N i,k (t) are blending functions V i are control points The matrix form of the uniform cubic B-spline curve is: V i V i (4.23) P i (t) = 1/6[t 3 t 2 t 1] V i V i+2 ME 165 Lecture Notes by R. B. Agarwal Computer Aided Design in Mechanical Engineering 4-21

### Geometric Modelling & Curves

Geometric Modelling & Curves Geometric Modeling Creating symbolic models of the physical world has long been a goal of mathematicians, scientists, engineers, etc. Recently technology has advanced sufficiently

### Computer Graphics. Geometric Modeling. Page 1. Copyright Gotsman, Elber, Barequet, Karni, Sheffer Computer Science - Technion. An Example.

An Example 2 3 4 Outline Objective: Develop methods and algorithms to mathematically model shape of real world objects Categories: Wire-Frame Representation Object is represented as as a set of points

### Figure 2.1: Center of mass of four points.

Chapter 2 Bézier curves are named after their inventor, Dr. Pierre Bézier. Bézier was an engineer with the Renault car company and set out in the early 196 s to develop a curve formulation which would

### Curves and Surfaces. Goals. How do we draw surfaces? How do we specify a surface? How do we approximate a surface?

Curves and Surfaces Parametric Representations Cubic Polynomial Forms Hermite Curves Bezier Curves and Surfaces [Angel 10.1-10.6] Goals How do we draw surfaces? Approximate with polygons Draw polygons

### We want to define smooth curves: - for defining paths of cameras or objects. - for defining 1D shapes of objects

lecture 10 - cubic curves - cubic splines - bicubic surfaces We want to define smooth curves: - for defining paths of cameras or objects - for defining 1D shapes of objects We want to define smooth surfaces

### (Refer Slide Time: 1:42)

Introduction to Computer Graphics Dr. Prem Kalra Department of Computer Science and Engineering Indian Institute of Technology, Delhi Lecture - 10 Curves So today we are going to have a new topic. So far

### 5. GEOMETRIC MODELING

5. GEOMETRIC MODELING Types of Curves and Their Mathematical Representation Types of Surfaces and Their Mathematical Representation Types of Solids and Their Mathematical Representation CAD/CAM Data Exchange

### parametric spline curves

parametric spline curves 1 curves used in many contexts fonts (2D) animation paths (3D) shape modeling (3D) different representation implicit curves parametric curves (mostly used) 2D and 3D curves are

### Computer Graphics CS 543 Lecture 12 (Part 1) Curves. Prof Emmanuel Agu. Computer Science Dept. Worcester Polytechnic Institute (WPI)

Computer Graphics CS 54 Lecture 1 (Part 1) Curves Prof Emmanuel Agu Computer Science Dept. Worcester Polytechnic Institute (WPI) So Far Dealt with straight lines and flat surfaces Real world objects include

### Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks

Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson

On-Line Geometric Modeling Notes QUADRATIC BÉZIER CURVES Kenneth I. Joy Visualization and Graphics Research Group Department of Computer Science University of California, Davis Overview The Bézier curve

### Essential Mathematics for Computer Graphics fast

John Vince Essential Mathematics for Computer Graphics fast Springer Contents 1. MATHEMATICS 1 Is mathematics difficult? 3 Who should read this book? 4 Aims and objectives of this book 4 Assumptions made

### Comal Independent School District Pre-AP Pre-Calculus Scope and Sequence

Comal Independent School District Pre- Pre-Calculus Scope and Sequence Third Quarter Assurances. The student will plot points in the Cartesian plane, use the distance formula to find the distance between

### Able Enrichment Centre - Prep Level Curriculum

Able Enrichment Centre - Prep Level Curriculum Unit 1: Number Systems Number Line Converting expanded form into standard form or vice versa. Define: Prime Number, Natural Number, Integer, Rational Number,

### 6.1 Application of Solid Models In mechanical engineering, a solid model is used for the following applications:

CHAPTER 6 SOLID MODELING 6.1 Application of Solid Models In mechanical engineering, a solid model is used for the following applications: 1. Graphics: generating drawings, surface and solid models 2. Design:

### TWO-DIMENSIONAL TRANSFORMATION

CHAPTER 2 TWO-DIMENSIONAL TRANSFORMATION 2.1 Introduction As stated earlier, Computer Aided Design consists of three components, namely, Design (Geometric Modeling), Analysis (FEA, etc), and Visualization

### the points are called control points approximating curve

Chapter 4 Spline Curves A spline curve is a mathematical representation for which it is easy to build an interface that will allow a user to design and control the shape of complex curves and surfaces.

### Engineering Geometry

Engineering Geometry Objectives Describe the importance of engineering geometry in design process. Describe coordinate geometry and coordinate systems and apply them to CAD. Review the right-hand rule.

### CHAPTER 1 Splines and B-splines an Introduction

CHAPTER 1 Splines and B-splines an Introduction In this first chapter, we consider the following fundamental problem: Given a set of points in the plane, determine a smooth curve that approximates the

### REVISED GCSE Scheme of Work Mathematics Higher Unit 6. For First Teaching September 2010 For First Examination Summer 2011 This Unit Summer 2012

REVISED GCSE Scheme of Work Mathematics Higher Unit 6 For First Teaching September 2010 For First Examination Summer 2011 This Unit Summer 2012 Version 1: 28 April 10 Version 1: 28 April 10 Unit T6 Unit

### CSE 167: Lecture 13: Bézier Curves. Jürgen P. Schulze, Ph.D. University of California, San Diego Fall Quarter 2011

CSE 167: Introduction to Computer Graphics Lecture 13: Bézier Curves Jürgen P. Schulze, Ph.D. University of California, San Diego Fall Quarter 2011 Announcements Homework project #6 due Friday, Nov 18

### Tangent and normal lines to conics

4.B. Tangent and normal lines to conics Apollonius work on conics includes a study of tangent and normal lines to these curves. The purpose of this document is to relate his approaches to the modern viewpoints

### Algebra 2 Year-at-a-Glance Leander ISD 2007-08. 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks

Algebra 2 Year-at-a-Glance Leander ISD 2007-08 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks Essential Unit of Study 6 weeks 3 weeks 3 weeks 6 weeks 3 weeks 3 weeks

### BEZIER CURVES AND SURFACES

Department of Applied Mathematics and Computational Sciences University of Cantabria UC-CAGD Group COMPUTER-AIDED GEOMETRIC DESIGN AND COMPUTER GRAPHICS: BEZIER CURVES AND SURFACES Andrés Iglesias e-mail:

### Section 1.1. Introduction to R n

The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to

### Modeling Curves and Surfaces

Modeling Curves and Surfaces Graphics I Modeling for Computer Graphics!? 1 How can we generate this kind of objects? Umm!? Mathematical Modeling! S Do not worry too much about your difficulties in mathematics,

### Overview Mathematical Practices Congruence

Overview Mathematical Practices Congruence 1. Make sense of problems and persevere in Experiment with transformations in the plane. solving them. Understand congruence in terms of rigid motions. 2. Reason

### An Overview of the Finite Element Analysis

CHAPTER 1 An Overview of the Finite Element Analysis 1.1 Introduction Finite element analysis (FEA) involves solution of engineering problems using computers. Engineering structures that have complex geometry

### pp. 4 8: Examples 1 6 Quick Check 1 6 Exercises 1, 2, 20, 42, 43, 64

Semester 1 Text: Chapter 1: Tools of Algebra Lesson 1-1: Properties of Real Numbers Day 1 Part 1: Graphing and Ordering Real Numbers Part 1: Graphing and Ordering Real Numbers Lesson 1-2: Algebraic Expressions

### arxiv:cs/ v1 [cs.gr] 22 Mar 2005

arxiv:cs/0503054v1 [cs.gr] 22 Mar 2005 ANALYTIC DEFINITION OF CURVES AND SURFACES BY PARABOLIC BLENDING by A.W. Overhauser Mathematical and Theoretical Sciences Department Scientific Laboratory, Ford Motor

### B2.53-R3: COMPUTER GRAPHICS. NOTE: 1. There are TWO PARTS in this Module/Paper. PART ONE contains FOUR questions and PART TWO contains FIVE questions.

B2.53-R3: COMPUTER GRAPHICS NOTE: 1. There are TWO PARTS in this Module/Paper. PART ONE contains FOUR questions and PART TWO contains FIVE questions. 2. PART ONE is to be answered in the TEAR-OFF ANSWER

### MATHEMATICS (CLASSES XI XII)

MATHEMATICS (CLASSES XI XII) General Guidelines (i) All concepts/identities must be illustrated by situational examples. (ii) The language of word problems must be clear, simple and unambiguous. (iii)

### Determine whether the following lines intersect, are parallel, or skew. L 1 : x = 6t y = 1 + 9t z = 3t. x = 1 + 2s y = 4 3s z = s

Homework Solutions 5/20 10.5.17 Determine whether the following lines intersect, are parallel, or skew. L 1 : L 2 : x = 6t y = 1 + 9t z = 3t x = 1 + 2s y = 4 3s z = s A vector parallel to L 1 is 6, 9,

### Unit Overview. Content Area: Math Unit Title: Functions and Their Graphs Target Course/Grade Level: Advanced Math Duration: 4 Weeks

Content Area: Math Unit Title: Functions and Their Graphs Target Course/Grade Level: Advanced Math Duration: 4 Weeks Unit Overview Description In this unit the students will examine groups of common functions

### COMPUTER AIDED GEOMETRIC DESIGN. Thomas W. Sederberg

COMPUTER AIDED GEOMETRIC DESIGN Thomas W. Sederberg September 28, 2016 ii T. W. Sederberg iii Preface I have taught a course at Brigham Young University titled Computer Aided Geometric Design since 1983.

### AQA Level 2 Certificate FURTHER MATHEMATICS

AQA Qualifications AQA Level 2 Certificate FURTHER MATHEMATICS Level 2 (8360) Our specification is published on our website (www.aqa.org.uk). We will let centres know in writing about any changes to the

### Algebra and Geometry Review (61 topics, no due date)

Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties

### Chapter 10: Topics in Analytic Geometry

Chapter 10: Topics in Analytic Geometry 10.1 Parabolas V In blue we see the parabola. It may be defined as the locus of points in the plane that a equidistant from a fixed point (F, the focus) and a fixed

### Copyrighted Material. Chapter 1 DEGREE OF A CURVE

Chapter 1 DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer two

### Estimated Pre Calculus Pacing Timeline

Estimated Pre Calculus Pacing Timeline 2010-2011 School Year The timeframes listed on this calendar are estimates based on a fifty-minute class period. You may need to adjust some of them from time to

### Working with Wireframe and Surface Design

Chapter 9 Working with Wireframe and Surface Design Learning Objectives After completing this chapter you will be able to: Create wireframe geometry. Create extruded surfaces. Create revolved surfaces.

### Roots and Coefficients of a Quadratic Equation Summary

Roots and Coefficients of a Quadratic Equation Summary For a quadratic equation with roots α and β: Sum of roots = α + β = and Product of roots = αβ = Symmetrical functions of α and β include: x = and

### H.Calculating Normal Vectors

Appendix H H.Calculating Normal Vectors This appendix describes how to calculate normal vectors for surfaces. You need to define normals to use the OpenGL lighting facility, which is described in Chapter

### Conic Sections in Cartesian and Polar Coordinates

Conic Sections in Cartesian and Polar Coordinates The conic sections are a family of curves in the plane which have the property in common that they represent all of the possible intersections of a plane

### The Not-Formula Book for C1

Not The Not-Formula Book for C1 Everything you need to know for Core 1 that won t be in the formula book Examination Board: AQA Brief This document is intended as an aid for revision. Although it includes

### The Essentials of CAGD

The Essentials of CAGD Chapter 2: Lines and Planes Gerald Farin & Dianne Hansford CRC Press, Taylor & Francis Group, An A K Peters Book www.farinhansford.com/books/essentials-cagd c 2000 Farin & Hansford

### Advanced Algebra 2. I. Equations and Inequalities

Advanced Algebra 2 I. Equations and Inequalities A. Real Numbers and Number Operations 6.A.5, 6.B.5, 7.C.5 1) Graph numbers on a number line 2) Order real numbers 3) Identify properties of real numbers

### Exam 1 Sample Question SOLUTIONS. y = 2x

Exam Sample Question SOLUTIONS. Eliminate the parameter to find a Cartesian equation for the curve: x e t, y e t. SOLUTION: You might look at the coordinates and notice that If you don t see it, we can

### Computer Animation: Art, Science and Criticism

Computer Animation: Art, Science and Criticism Tom Ellman Harry Roseman Lecture 4 Parametric Curve A procedure for distorting a straight line into a (possibly) curved line. The procedure lives in a black

### Parametric Curves. (Com S 477/577 Notes) Yan-Bin Jia. Oct 8, 2015

Parametric Curves (Com S 477/577 Notes) Yan-Bin Jia Oct 8, 2015 1 Introduction A curve in R 2 (or R 3 ) is a differentiable function α : [a,b] R 2 (or R 3 ). The initial point is α[a] and the final point

### Algebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions.

Chapter 1 Vocabulary identity - A statement that equates two equivalent expressions. verbal model- A word equation that represents a real-life problem. algebraic expression - An expression with variables.

### 3 Drawing 2D shapes. Launch form Z.

3 Drawing 2D shapes Launch form Z. If you have followed our instructions to this point, five icons will be displayed in the upper left corner of your screen. You can tear the three shown below off, to

### TRIGONOMETRY GRADES THE EWING PUBLIC SCHOOLS 2099 Pennington Road Ewing, NJ 08618

TRIGONOMETRY GRADES 11-12 THE EWING PUBLIC SCHOOLS 2099 Pennington Road Ewing, NJ 08618 Board Approval Date: October 29, 2012 Michael Nitti Written by: EHS Mathematics Department Superintendent In accordance

### Biggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress

Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation

### Common Curriculum Map. Discipline: Math Course: College Algebra

Common Curriculum Map Discipline: Math Course: College Algebra August/September: 6A.5 Perform additions, subtraction and multiplication of complex numbers and graph the results in the complex plane 8a.4a

### Understanding Basic Calculus

Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other

### Portable Assisted Study Sequence ALGEBRA IIA

SCOPE This course is divided into two semesters of study (A & B) comprised of five units each. Each unit teaches concepts and strategies recommended for intermediate algebra students. The first half of

### Higher Education Math Placement

Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication

### Volumes of Revolution

Mathematics Volumes of Revolution About this Lesson This lesson provides students with a physical method to visualize -dimensional solids and a specific procedure to sketch a solid of revolution. Students

### South Carolina College- and Career-Ready (SCCCR) Pre-Calculus

South Carolina College- and Career-Ready (SCCCR) Pre-Calculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know

### Solving Quadratic & Higher Degree Inequalities

Ch. 8 Solving Quadratic & Higher Degree Inequalities We solve quadratic and higher degree inequalities very much like we solve quadratic and higher degree equations. One method we often use to solve quadratic

16.810 Engineering Design and Rapid Prototyping Lecture 4 Computer Aided Design (CAD) Instructor(s) Prof. Olivier de Weck January 6, 2005 Plan for Today CAD Lecture (ca. 50 min) CAD History, Background

### MA 323 Geometric Modelling Course Notes: Day 02 Model Construction Problem

MA 323 Geometric Modelling Course Notes: Day 02 Model Construction Problem David L. Finn November 30th, 2004 In the next few days, we will introduce some of the basic problems in geometric modelling, and

### Birmingham City Schools

Activity 1 Classroom Rules & Regulations Policies & Procedures Course Curriculum / Syllabus LTF Activity: Interval Notation (Precal) 2 Pre-Assessment 3 & 4 1.2 Functions and Their Properties 5 LTF Activity:

### Solutions for Review Problems

olutions for Review Problems 1. Let be the triangle with vertices A (,, ), B (4,, 1) and C (,, 1). (a) Find the cosine of the angle BAC at vertex A. (b) Find the area of the triangle ABC. (c) Find a vector

### HIGH SCHOOL: GEOMETRY (Page 1 of 4)

HIGH SCHOOL: GEOMETRY (Page 1 of 4) Geometry is a complete college preparatory course of plane and solid geometry. It is recommended that there be a strand of algebra review woven throughout the course

### 3.5 Spline interpolation

3.5 Spline interpolation Given a tabulated function f k = f(x k ), k = 0,... N, a spline is a polynomial between each pair of tabulated points, but one whose coefficients are determined slightly non-locally.

### Parametric Representation of Curves and Surfaces

Parametric Representation of Crves and Srfaces How does the compter compte crves and srfaces? MAE 455 Compter-Aided Design and Drafting Types of Crve Eqations Implicit Form Explicit Form Parametric Form

### www.mathsbox.org.uk ab = c a If the coefficients a,b and c are real then either α and β are real or α and β are complex conjugates

Further Pure Summary Notes. Roots of Quadratic Equations For a quadratic equation ax + bx + c = 0 with roots α and β Sum of the roots Product of roots a + b = b a ab = c a If the coefficients a,b and c

### Introduction to Computer Graphics Marie-Paule Cani & Estelle Duveau

Introduction to Computer Graphics Marie-Paule Cani & Estelle Duveau 04/02 Introduction & projective rendering 11/02 Prodedural modeling, Interactive modeling with parametric surfaces 25/02 Introduction

16.810 Engineering Design and Rapid Prototyping Lecture 3a Computer Aided Design (CAD) Instructor(s) Prof. Olivier de Weck January 16, 2007 Plan for Today CAD Lecture (ca. 50 min) CAD History, Background

### Section 2.1 Rectangular Coordinate Systems

P a g e 1 Section 2.1 Rectangular Coordinate Systems 1. Pythagorean Theorem In a right triangle, the lengths of the sides are related by the equation where a and b are the lengths of the legs and c is

### The Australian Curriculum Mathematics

The Australian Curriculum Mathematics Mathematics ACARA The Australian Curriculum Number Algebra Number place value Fractions decimals Real numbers Foundation Year Year 1 Year 2 Year 3 Year 4 Year 5 Year

### Student Performance Q&A:

Student Performance Q&A: AP Calculus AB and Calculus BC Free-Response Questions The following comments on the free-response questions for AP Calculus AB and Calculus BC were written by the Chief Reader,

### Interactive Math Glossary Terms and Definitions

Terms and Definitions Absolute Value the magnitude of a number, or the distance from 0 on a real number line Additive Property of Area the process of finding an the area of a shape by totaling the areas

### Chapter 3-3D Modeling

Chapter 3-3D Modeling Polygon Meshes Geometric Primitives Interpolation Curves Levels Of Detail (LOD) Constructive Solid Geometry (CSG) Extrusion & Rotation Volume- and Point-based Graphics 1 The 3D rendering

### The degree of the polynomial function is n. We call the term the leading term, and is called the leading coefficient. 0 =

Math 1310 Section 4.1: Polynomial Functions and Their Graphs A polynomial function is a function of the form = + + +...+ + where 0,,,, are real numbers and n is a whole number. The degree of the polynomial

### Functions and Equations

Centre for Education in Mathematics and Computing Euclid eworkshop # Functions and Equations c 014 UNIVERSITY OF WATERLOO Euclid eworkshop # TOOLKIT Parabolas The quadratic f(x) = ax + bx + c (with a,b,c

### String Art Mathematics: An Introduction to Geometry Expressions and Math Illustrations Stephen Arnold Compass Learning Technologies

String Art Mathematics: An Introduction to Geometry Expressions and Math Illustrations Stephen Arnold Compass Learning Technologies Introduction How do you create string art on a computer? In this lesson

### MyMathLab ecourse for Developmental Mathematics

MyMathLab ecourse for Developmental Mathematics, North Shore Community College, University of New Orleans, Orange Coast College, Normandale Community College Table of Contents Module 1: Whole Numbers and

### Content. Chapter 4 Functions 61 4.1 Basic concepts on real functions 62. Credits 11

Content Credits 11 Chapter 1 Arithmetic Refresher 13 1.1 Algebra 14 Real Numbers 14 Real Polynomials 19 1.2 Equations in one variable 21 Linear Equations 21 Quadratic Equations 22 1.3 Exercises 28 Chapter

### 2 Topics in 3D Geometry

2 Topics in 3D Geometry In two dimensional space, we can graph curves and lines. In three dimensional space, there is so much extra space that we can graph planes and surfaces in addition to lines and

### Introduction. The Aims & Objectives of the Mathematical Portion of the IBA Entry Test

Introduction The career world is competitive. The competition and the opportunities in the career world become a serious problem for students if they do not do well in Mathematics, because then they are

### Algebra II Semester Exam Review Sheet

Name: Class: Date: ID: A Algebra II Semester Exam Review Sheet 1. Translate the point (2, 3) left 2 units and up 3 units. Give the coordinates of the translated point. 2. Use a table to translate the graph

### Graphical objects geometrical data handling inside DWG file

Graphical objects geometrical data handling inside DWG file Author: Vidmantas Nenorta, Kaunas University of Technology, Lithuania, vidmantas.nenorta@ktu.lt Abstract The paper deals with required (necessary)

### Homework 3 Model Solution Section

Homework 3 Model Solution Section 12.6 13.1. 12.6.3 Describe and sketch the surface + z 2 = 1. If we cut the surface by a plane y = k which is parallel to xz-plane, the intersection is + z 2 = 1 on a plane,

### since by using a computer we are limited to the use of elementary arithmetic operations

> 4. Interpolation and Approximation Most functions cannot be evaluated exactly: x, e x, ln x, trigonometric functions since by using a computer we are limited to the use of elementary arithmetic operations

### 2D Parametric Design. Chapter. Objectives

Chapter 1 2D Parametric Design In this chapter, you learn about the benefits and characteristics of creating parametric 2D geometry and how various types of dynamic constraints can be assigned to geometry

### (a) We have x = 3 + 2t, y = 2 t, z = 6 so solving for t we get the symmetric equations. x 3 2. = 2 y, z = 6. t 2 2t + 1 = 0,

Name: Solutions to Practice Final. Consider the line r(t) = 3 + t, t, 6. (a) Find symmetric equations for this line. (b) Find the point where the first line r(t) intersects the surface z = x + y. (a) We

### EL-9650/9600c/9450/9400 Handbook Vol. 1

Graphing Calculator EL-9650/9600c/9450/9400 Handbook Vol. Algebra EL-9650 EL-9450 Contents. Linear Equations - Slope and Intercept of Linear Equations -2 Parallel and Perpendicular Lines 2. Quadratic Equations

### The Locus of the Focus of a Rolling Parabola

The Locus of the Focus of a Rolling Parabola Anurag Agarwal & James Marengo October 6, 2009 1 Introduction The catenary can easily be mistaken for a parabola. Even Galileo made this error. In his Discorsi

### Calculating Areas Section 6.1

A B I L E N E C H R I S T I A N U N I V E R S I T Y Department of Mathematics Calculating Areas Section 6.1 Dr. John Ehrke Department of Mathematics Fall 2012 Measuring Area By Slicing We first defined

### The Parabola. By: OpenStaxCollege

The Parabola By: OpenStaxCollege The Olympic torch concludes its journey around the world when it is used to light the Olympic cauldron during the opening ceremony. (credit: Ken Hackman, U.S. Air Force)

### REVISED GCSE Scheme of Work Mathematics Higher Unit T3. For First Teaching September 2010 For First Examination Summer 2011

REVISED GCSE Scheme of Work Mathematics Higher Unit T3 For First Teaching September 2010 For First Examination Summer 2011 Version 1: 28 April 10 Version 1: 28 April 10 Unit T3 Unit T3 This is a working

### 4. Factor polynomials over complex numbers, describe geometrically, and apply to real-world situations. 5. Determine and apply relationships among syn

I The Real and Complex Number Systems 1. Identify subsets of complex numbers, and compare their structural characteristics. 2. Compare and contrast the properties of real numbers with the properties of

### Dhiren Bhatia Carnegie Mellon University

Dhiren Bhatia Carnegie Mellon University University Course Evaluations available online Please Fill! December 4 : In-class final exam Held during class time All students expected to give final this date

Academic Content Standards Grade Eight and Grade Nine Ohio Algebra 1 2008 Grade Eight STANDARDS Number, Number Sense and Operations Standard Number and Number Systems 1. Use scientific notation to express