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

Size: px
Start display at page:

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

## Transcription

1 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 some very simple methods for solving them. Each of the motivating problems has different flavor depending on the type of object that is being created. For our purposes, there are two types of objects to model; curves and surfaces. One can consider solids as a different class of objects, but since a solid is bounded by a surface or a collection of surfaces, for our purposes in this course a solid is represented by its bounding surfaces. In this chapter, we will pose the motivating problems in generic terms, but concentrate on the version of the problems for curves when we discuss the solution techniques. The generalization of the motivating problems to surfaces is straightforward and simple, but the solution methods are more complex to state in mathematical terms. 2.1 The Model Construction Problem The first and most basic problem in geometric modelling is to construct a model of an object by specifying geometric primitives. In this problem, the modeller needs to supply simple geometric information (points, lines, planes, etc) in order to obtain a model of the object. The solution method takes the geometric primitives and constructs the model. The solution method is normally algorithmic and can be modified at the modeller s discretion, depending on the exact problem that needs to be solved. The solution methods to the model construction problem generally are described by the type of objects that they generate. For instance, in the subsections that follow, we describe two types of curve constructions; piecewise linear curves and piecewise circular curves. There are of course other curve construction methods that are substantially more complicated to describe. In general, these more complicated methods are better modelling methods. We will discuss some of these other methods later in the course, but piecewise linear curves and piecewise circular curves suffice for a discussion of the construction problem and some of the important issues in the construction problem. It is worth repeating that the solution method to the construction problem greatly affects the solutions to any other problem that the modeller may want to tackle. This is due to the fact that construction problem specifies the type of model and therefore the properties of the model. The properties the model possesses will of course restrict the ability of the model to solve other problems and affect the analysis of the model. Sometimes the model s

2 2-2 ability to solve the problem is hampered by the construction method. In fact, one practical concern that we will not discuss that often is the conversion of one model to another model. 2.2 Solution by Piecewise Linear Curves The simplest solution to the model construction problem for curves is to construct a piecewise linear curve to approximate a curve. A piecewise linear curve is just a finite sequence of line segments. To specify a piecewise linear curve, one only needs to specify a sequence of points {p 0, p 1,, p m }, and then form the line segments p i p i+1. Thus, we form m line segments from the m + 1 points. This provides a very simple geometric construction to approximate a drawn curve in a plane, see figure below. The actual curve constructed in this manner is sometimes called a polyline and the function that describes the curve is a piecewise linear interpolant or piecewise linear function. Figure 1: A piecewise linear curve It is a basic but deep result in mathematical analysis that any continuous curve can be approximated closely in distance arbitrarily well with a piecewise linear curve. This is a direct result of the ɛ-δ definition of continuity. A significant problem with this basic result is that it does not say how to arrange the points and even how many points are needed. It simply says that it is possible to approximate any continuous curve with a piecewise linear curve. Nevertheless, piecewise linear curves and piecewise linear function are the fundamental building blocks of every graphics engine and every approximation scheme. The reason for this is simple. The pixels on the screen can be viewed as points, and to create a curve, we need to it is easy to sample points on the curve and connect them by straight lines. Moreover, digital computers can only add and multiply, which are the fundamental operations of linear approximations. Figure 2: Increasing accuracy of approximation with more data points

3 2-3 There are several obvious problems with using piecewise linear curves to provide a good solution to the construction problem. A serious problem is that to obtain a good approximation with a piecewise linear curves one might need to supply a large number of points, and consequently a large number of line segments. A second problem is that the model generated by piecewise linear curves is not smooth. There is a noticeable angle at each point, when only a few control points are chosen. When more points are used the angle becomes less discernable, but this only highlights the serious problem of obtaining a good approximation. It is worth mentioning that despite the above problems with piecewise linear curves, they are an important tool in geometric modelling. One reason for this is that most of the other methods will eventually use a piecewise linear curve using a high number of points as an approximation of the model curve that it will generate. The modeller does not have to generate all the points themselves, the modelling method typical generates most of the points from a few select points generated by the modeller. The reason for this is simple. The methods for generating geometric models are limited by the discrete nature of computers and the resolution of the screen and computer control. To connect between discrete points that are very close together, most computer systems will use a straight line for motion and as most problems are continuous and differentiable, calculus ensures that locally curves can be approximated by straight lines. 2.3 Solution by Piecewise Circular Curves An alternate solution method to piecewise linear curves that can remove the smoothness problem in the model curve is to use piecewise circular curves. The difference between piecewise circular curves and piecewise linear curves is to replace the line segments with circular arcs. This will be a fundamental tool in geometric modelling to use basic curve elements, and joining them together to create a model. Figure 3: A nonsmooth piecewise circular curve To construct a piecewise circular curve, recall that it is possible to construct a circular arc that passes through three noncollinear points, see below for the construction. Therefore, by providing a sequence of points {p 0, p 1,, p n } with n = 2m an even number, one can construct a circular curve by finding the circle that passes through each set of three points p 2k, p 2k+1 and p 2k+2 with k = 0, 1,, n 1 and then finding the circular arc that goes from p 2k to p 2k+2 and passes through p 2k+1, see diagram above. We note it is not necessary to require that the points p 2k, p 2k+1 and p 2k+2 to be noncollinear. The case of three collinear points is covered by noticing in the construction of a circle through three noncollinear points as the the points get closer to being collinear the circle produced approaches a straight line. Therefore, for three collinear points the circle through the three points is a straight line. The circular arc is then the line segment p 2k p 2k+2. Note the circular arc makes sense

4 2-4 (at least classically) when p 2k+1 is between, p 2k and p 2k+2, though we will see later that it makes sense projectively when p 2k+1 is not between p 2k and p 2k+2. GEOMETRIC CONSTRUCTION: The construction of a circle through three noncollinear points. Let A, B, C be three given noncollinear points. To construct a circle through these three points, we first construct the triangle with vertices A, B, C. Then we construct the circumscribing circle for this triangle. This is done by first finding the midpoints of the sides of the triangle, called these A, B, C with A on side BC, B on side AC and C on side AB. Construct lines perpendicular to AB through C, perpendicular to AC through B, and perpendicular to BC through A. These lines all intersect at a common point Q called the circumcenter of the triangle. This point Q is the center of the circumscribed circle to the triangle ABC. To see this note that the Pythagorean theorem applied to the right triangles AC Q and BC Q (since C Q is in both triangles and dist(ac ) = dist(bc ) shows that the distance dist(aq) = dist(bq). Applying the same argument to the pair of triangles BA Q and CA Q yields that dist(bq) = dist(cq). Therefore the circle of radius dist(aq) through Q passes through A, B, C. B B' A' A C' Q C Figure 4: Construction of a Circle Through Three Points It is not hard to modify this construction of a circle through three noncollinear points, to obtain a circular arc that starts at point A and ends at point C and includes point B. All that one needs to do is construct the lines AB and BC, then construct the midpoints and the perpendicular bisectors of these lines. The point of intersubsection is the center of the circlular arc. Then draw the arc. In the construction of a (nonsmooth) piecewise circular curve above, we construct m circular arcs from the 2m + 1 given points. We note this construction only generates a continuous curve. At the points p 2i with i = 1, 2,, m 1 there is generically a discernible angle. [Generically is the mathematically terminology that means usually or more accurately unless unusual circumstances prevail.] This means that without quanitifying measures that if one randomly chooses the points p i there will be a discernible angle almost always at the points p 2i. This is not aesthetically pleasing or really practical. We have only replaced the straight lines with circles, at the inconvenience of having to supply more points.

5 2-5 The advantage of piecewise circular curves is that one may with the appropriate choice of the points {p i } obtain a smooth model. It is important that the appropriate choice of the points follow simple rules that is easy to compute. To develop these rules, we make the following observations. For a piecewise circular curve to be smooth, we need the tangent lines at the joint points p 2k (k = 1, 2,, m 1) as computed from each circular arc to be identical. Figure 5: Illustration of Properties of Smooth Piecewise Curves The fact that the tangent line to a circle is perpendicular to the radial line implies that radial lines (line between the center and a point on the circle) at p 2k must be the same for each circular arc. This means that centers of consecutive circular arcs must be collinear with the joint point that the arcs share in common. Figure 6: Illustration of Properties of Smooth Piecewise Curves It is possible to construct a piecewise circular curve consisting of two circular arcs by giving the three points p 0, p 2 and p 4 and the tangent line at p 0. (see ASIDE: Construction of circle from two points A and B and a tangent line l at A) Figure 7: Construction with a Tangent Line

6 2-6 Generalizing the observation, one can construct a smooth circular arc consisting of m arcs by giving m + 1 points p 0, p 1, p 2,, p m and a tangent line l at p 0. First construct the circular arc starting at p 0 and ending at p 1 using the tangent line l. Construct the tangent line at p 1 from this first arc, and use that tangent line to construct the circular arc from p 1 to p 2. Figure 8: Construction of a Smooth Circular Curve The basic construction that allows one to construct a smooth piecewise circular curve requires a slight modification of the construction of a circle through three noncollinear points. In this construction, we want to construct a circle given two points and a tangent line at a given line. Then, a slight modification of the construction will give the desired circular arc. GEOMETRIC CONSTRUCTION: Construction of circle from two points A and B and a tangent line l at A: In this construction, there are two cases to consider. The first case is when B lies on the line l. In this case, the circular arc is the line segment AB. The second case is when B does not lie on the line l. The circle is constructed as follows. Figure 9: Construction of a Circle from Two Points and Tangent Line Construct a line l perpendicular to l passing through A. (This is the radial line of the circular arcs that pass through A).

7 2-7 Construct the perpendicular bisector m of the line segment AB. Let C be the intersubsection of the lines l and m. This is the center of the circular arc. The lengths AC and BC are equal because of the Pythagorean theorem. To construct a circular arc, given a tangent line l at A, one proceeds as above to construct the circle, but in addition one must choose or be given a direction on the line (a tangent vector at A). This allows to choose which arc of the circle goes from A to B. We choose the arc leaving the point A going towards the point B in the direction chosen. Notice that for the collinear case, the only orientation that makes sense classically is the direction that points from A towards B. However, in either case, there is an implied tangent vector for the circular arc at point B given by the vector heading in the direction that the arc continues, see diagram below. This allows us to continue the construction from B by adding a new circular arc, and thus obtaining a smooth circular curve. The construction of a smooth piecewise circular curve using the construction of a circle from two points and a tangent line allows us to obtain a globally smooth curve. Notice that for this construction, we need to supply m + 1 points (the end points of each circular arc) and a tangent line at one of the points, and moreover given any m + 1 points there is a oneparameter family (defined by a choice of a tangent line) of smooth piecewise circular curves. An alternate method is to use the first three consecutive points (or any three consecutive points) to define a circular arc and then base the rest of the circular constructions off this circle, as the construction of one circle defines the the necessary tangent line for the remainder of the circles. Exercises 1. (Computational) Approximate the curve below with a piecewise linear curve using four line segments, five line segments, ten line segments. How many line segments are needed to produce a good approximation? How should we place the control points determining the line segments? Figure 10: Approximate this Curve with a Piecewise Linear Curve 2. (Thought) Does the construction for a piecewise linear curve require the points to be in a plane? Can you construct a piecewise linear curve with the control points in three dimensional space? What needs to be modified from the planar case?

8 (Computational) Approximate the curve below with a smooth piecewise circular curve with four arcs, five arcs, ten arcs. Is it possible to produce a good approximation with a piecewise circular curve? (Why or why not?) If so how many circular arcs are needed to produce a good approximation? How should we place the control points determining the circular arcs? Figure 11: Approximate this Curve with a Piecewise Circular Curve 4. (Interactive) Complete the interactive exercises associated with the applet on the course web-page Piecewise Circular Curves 5. (Thought) Does the construction for a piecewise circular curve require the points to be in a plane? Can you construct a piecewise circular curve with control points in three-dimensional space? What if anything needs to be modified from the planar case?

### DEFINITIONS. Perpendicular Two lines are called perpendicular if they form a right angle.

DEFINITIONS Degree A degree is the 1 th part of a straight angle. 180 Right Angle A 90 angle is called a right angle. Perpendicular Two lines are called perpendicular if they form a right angle. Congruent

### Inversion. Chapter 7. 7.1 Constructing The Inverse of a Point: If P is inside the circle of inversion: (See Figure 7.1)

Chapter 7 Inversion Goal: In this chapter we define inversion, give constructions for inverses of points both inside and outside the circle of inversion, and show how inversion could be done using Geometer

### Chapter 6 Notes: Circles

Chapter 6 Notes: Circles IMPORTANT TERMS AND DEFINITIONS A circle is the set of all points in a plane that are at a fixed distance from a given point known as the center of the circle. Any line segment

### Chapters 6 and 7 Notes: Circles, Locus and Concurrence

Chapters 6 and 7 Notes: Circles, Locus and Concurrence IMPORTANT TERMS AND DEFINITIONS A circle is the set of all points in a plane that are at a fixed distance from a given point known as the center of

### Definitions, Postulates and Theorems

Definitions, s and s Name: Definitions Complementary Angles Two angles whose measures have a sum of 90 o Supplementary Angles Two angles whose measures have a sum of 180 o A statement that can be proven

### 1. A student followed the given steps below to complete a construction. Which type of construction is best represented by the steps given above?

1. A student followed the given steps below to complete a construction. Step 1: Place the compass on one endpoint of the line segment. Step 2: Extend the compass from the chosen endpoint so that the width

### Selected practice exam solutions (part 5, item 2) (MAT 360)

Selected practice exam solutions (part 5, item ) (MAT 360) Harder 8,91,9,94(smaller should be replaced by greater )95,103,109,140,160,(178,179,180,181 this is really one problem),188,193,194,195 8. On

### Circle Name: Radius: Diameter: Chord: Secant:

12.1: Tangent Lines Congruent Circles: circles that have the same radius length Diagram of Examples Center of Circle: Circle Name: Radius: Diameter: Chord: Secant: Tangent to A Circle: a line in the plane

### Geometry Chapter 1. 1.1 Point (pt) 1.1 Coplanar (1.1) 1.1 Space (1.1) 1.2 Line Segment (seg) 1.2 Measure of a Segment

Geometry Chapter 1 Section Term 1.1 Point (pt) Definition A location. It is drawn as a dot, and named with a capital letter. It has no shape or size. undefined term 1.1 Line A line is made up of points

### Geometry: Unit 1 Vocabulary TERM DEFINITION GEOMETRIC FIGURE. Cannot be defined by using other figures.

Geometry: Unit 1 Vocabulary 1.1 Undefined terms Cannot be defined by using other figures. Point A specific location. It has no dimension and is represented by a dot. Line Plane A connected straight path.

### Curriculum Map by Block Geometry Mapping for Math Block Testing 2007-2008. August 20 to August 24 Review concepts from previous grades.

Curriculum Map by Geometry Mapping for Math Testing 2007-2008 Pre- s 1 August 20 to August 24 Review concepts from previous grades. August 27 to September 28 (Assessment to be completed by September 28)

### Chapter 3.1 Angles. Geometry. Objectives: Define what an angle is. Define the parts of an angle.

Chapter 3.1 Angles Define what an angle is. Define the parts of an angle. Recall our definition for a ray. A ray is a line segment with a definite starting point and extends into infinity in only one direction.

### 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

### GEOMETRY. Chapter 1: Foundations for Geometry. Name: Teacher: Pd:

GEOMETRY Chapter 1: Foundations for Geometry Name: Teacher: Pd: Table of Contents Lesson 1.1: SWBAT: Identify, name, and draw points, lines, segments, rays, and planes. Pgs: 1-4 Lesson 1.2: SWBAT: Use

### Straight Line. Paper 1 Section A. O xy

PSf Straight Line Paper 1 Section A Each correct answer in this section is worth two marks. 1. The line with equation = a + 4 is perpendicular to the line with equation 3 + + 1 = 0. What is the value of

### Lesson 2: Circles, Chords, Diameters, and Their Relationships

Circles, Chords, Diameters, and Their Relationships Student Outcomes Identify the relationships between the diameters of a circle and other chords of the circle. Lesson Notes Students are asked to construct

### Visualizing Triangle Centers Using Geogebra

Visualizing Triangle Centers Using Geogebra Sanjay Gulati Shri Shankaracharya Vidyalaya, Hudco, Bhilai India http://mathematicsbhilai.blogspot.com/ sanjaybhil@gmail.com ABSTRACT. In this paper, we will

### Lesson 5-3: Concurrent Lines, Medians and Altitudes

Playing with bisectors Yesterday we learned some properties of perpendicular bisectors of the sides of triangles, and of triangle angle bisectors. Today we are going to use those skills to construct special

### Factoring Patterns in the Gaussian Plane

Factoring Patterns in the Gaussian Plane Steve Phelps Introduction This paper describes discoveries made at the Park City Mathematics Institute, 00, as well as some proofs. Before the summer I understood

### Grade 7 & 8 Math Circles Circles, Circles, Circles March 19/20, 2013

Faculty of Mathematics Waterloo, Ontario N2L 3G Introduction Grade 7 & 8 Math Circles Circles, Circles, Circles March 9/20, 203 The circle is a very important shape. In fact of all shapes, the circle is

### GEOMETRY CONCEPT MAP. Suggested Sequence:

CONCEPT MAP GEOMETRY August 2011 Suggested Sequence: 1. Tools of Geometry 2. Reasoning and Proof 3. Parallel and Perpendicular Lines 4. Congruent Triangles 5. Relationships Within Triangles 6. Polygons

### Conjectures. Chapter 2. Chapter 3

Conjectures Chapter 2 C-1 Linear Pair Conjecture If two angles form a linear pair, then the measures of the angles add up to 180. (Lesson 2.5) C-2 Vertical Angles Conjecture If two angles are vertical

### Geometry Course Summary Department: Math. Semester 1

Geometry Course Summary Department: Math Semester 1 Learning Objective #1 Geometry Basics Targets to Meet Learning Objective #1 Use inductive reasoning to make conclusions about mathematical patterns Give

### 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

### The Inversion Transformation

The Inversion Transformation A non-linear transformation The transformations of the Euclidean plane that we have studied so far have all had the property that lines have been mapped to lines. Transformations

### Angles that are between parallel lines, but on opposite sides of a transversal.

GLOSSARY Appendix A Appendix A: Glossary Acute Angle An angle that measures less than 90. Acute Triangle Alternate Angles A triangle that has three acute angles. Angles that are between parallel lines,

### Sample Test Questions

mathematics College Algebra Geometry Trigonometry Sample Test Questions A Guide for Students and Parents act.org/compass Note to Students Welcome to the ACT Compass Sample Mathematics Test! You are about

### POTENTIAL REASONS: Definition of Congruence:

Sec 6 CC Geometry Triangle Pros Name: POTENTIAL REASONS: Definition Congruence: Having the exact same size and shape and there by having the exact same measures. Definition Midpoint: The point that divides

### CIRCLE COORDINATE GEOMETRY

CIRCLE COORDINATE GEOMETRY (EXAM QUESTIONS) Question 1 (**) A circle has equation x + y = 2x + 8 Determine the radius and the coordinates of the centre of the circle. r = 3, ( 1,0 ) Question 2 (**) A circle

### Unit 3: Circles and Volume

Unit 3: Circles and Volume This unit investigates the properties of circles and addresses finding the volume of solids. Properties of circles are used to solve problems involving arcs, angles, sectors,

### ME 111: Engineering Drawing

ME 111: Engineering Drawing Lecture # 14 (10/10/2011) Development of Surfaces http://www.iitg.ernet.in/arindam.dey/me111.htm http://www.iitg.ernet.in/rkbc/me111.htm http://shilloi.iitg.ernet.in/~psr/ Indian

### A Short Introduction to Computer Graphics

A Short Introduction to Computer Graphics Frédo Durand MIT Laboratory for Computer Science 1 Introduction Chapter I: Basics Although computer graphics is a vast field that encompasses almost any graphical

### GEOMETRY. Constructions OBJECTIVE #: G.CO.12

GEOMETRY Constructions OBJECTIVE #: G.CO.12 OBJECTIVE Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic

### Contents. 2 Lines and Circles 3 2.1 Cartesian Coordinates... 3 2.2 Distance and Midpoint Formulas... 3 2.3 Lines... 3 2.4 Circles...

Contents Lines and Circles 3.1 Cartesian Coordinates.......................... 3. Distance and Midpoint Formulas.................... 3.3 Lines.................................. 3.4 Circles..................................

### Incenter Circumcenter

TRIANGLE: Centers: Incenter Incenter is the center of the inscribed circle (incircle) of the triangle, it is the point of intersection of the angle bisectors of the triangle. The radius of incircle is

### THREE DIMENSIONAL GEOMETRY

Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,

### Gymnázium, Brno, Slovanské nám. 7, SCHEME OF WORK Mathematics SCHEME OF WORK. http://agb.gymnaslo. cz

SCHEME OF WORK Subject: Mathematics Year: Third grade, 3.X School year:../ List of topics Topics Time period 1. Revision (functions, plane geometry) September 2. Constructive geometry in the plane October

### CHAPTER FIVE. 5. Equations of Lines in R 3

118 CHAPTER FIVE 5. Equations of Lines in R 3 In this chapter it is going to be very important to distinguish clearly between points and vectors. Frequently in the past the distinction has only been a

### Mathematics Geometry Unit 1 (SAMPLE)

Review the Geometry sample year-long scope and sequence associated with this unit plan. Mathematics Possible time frame: Unit 1: Introduction to Geometric Concepts, Construction, and Proof 14 days This

### Solving Simultaneous Equations and Matrices

Solving Simultaneous Equations and Matrices The following represents a systematic investigation for the steps used to solve two simultaneous linear equations in two unknowns. The motivation for considering

### Mathematics 3301-001 Spring 2015 Dr. Alexandra Shlapentokh Guide #3

Mathematics 3301-001 Spring 2015 Dr. Alexandra Shlapentokh Guide #3 The problems in bold are the problems for Test #3. As before, you are allowed to use statements above and all postulates in the proofs

### Section 1.4. Lines, Planes, and Hyperplanes. The Calculus of Functions of Several Variables

The Calculus of Functions of Several Variables Section 1.4 Lines, Planes, Hyperplanes In this section we will add to our basic geometric understing of R n by studying lines planes. If we do this carefully,

### CSU Fresno Problem Solving Session. Geometry, 17 March 2012

CSU Fresno Problem Solving Session Problem Solving Sessions website: http://zimmer.csufresno.edu/ mnogin/mfd-prep.html Math Field Day date: Saturday, April 21, 2012 Math Field Day website: http://www.csufresno.edu/math/news

### Chapter 3. Inversion and Applications to Ptolemy and Euler

Chapter 3. Inversion and Applications to Ptolemy and Euler 2 Power of a point with respect to a circle Let A be a point and C a circle (Figure 1). If A is outside C and T is a point of contact of a tangent

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

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

### INCIDENCE-BETWEENNESS GEOMETRY

INCIDENCE-BETWEENNESS GEOMETRY MATH 410, CSUSM. SPRING 2008. PROFESSOR AITKEN This document covers the geometry that can be developed with just the axioms related to incidence and betweenness. The full

### 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

### Lesson 18: Looking More Carefully at Parallel Lines

Student Outcomes Students learn to construct a line parallel to a given line through a point not on that line using a rotation by 180. They learn how to prove the alternate interior angles theorem using

### Solutions to old Exam 1 problems

Solutions to old Exam 1 problems Hi students! I am putting this old version of my review for the first midterm review, place and time to be announced. Check for updates on the web site as to which sections

### PCHS ALGEBRA PLACEMENT TEST

MATHEMATICS Students must pass all math courses with a C or better to advance to the next math level. Only classes passed with a C or better will count towards meeting college entrance requirements. If

### alternate interior angles

alternate interior angles two non-adjacent angles that lie on the opposite sides of a transversal between two lines that the transversal intersects (a description of the location of the angles); alternate

### 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

### L 2 : x = s + 1, y = s, z = 4s + 4. 3. Suppose that C has coordinates (x, y, z). Then from the vector equality AC = BD, one has

The line L through the points A and B is parallel to the vector AB = 3, 2, and has parametric equations x = 3t + 2, y = 2t +, z = t Therefore, the intersection point of the line with the plane should satisfy:

### 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

### New York State Student Learning Objective: Regents Geometry

New York State Student Learning Objective: Regents Geometry All SLOs MUST include the following basic components: Population These are the students assigned to the course section(s) in this SLO all students

### Solutions to Practice Problems

Higher Geometry Final Exam Tues Dec 11, 5-7:30 pm Practice Problems (1) Know the following definitions, statements of theorems, properties from the notes: congruent, triangle, quadrilateral, isosceles

### Example SECTION 13-1. X-AXIS - the horizontal number line. Y-AXIS - the vertical number line ORIGIN - the point where the x-axis and y-axis cross

CHAPTER 13 SECTION 13-1 Geometry and Algebra The Distance Formula COORDINATE PLANE consists of two perpendicular number lines, dividing the plane into four regions called quadrants X-AXIS - the horizontal

### Mathematics Notes for Class 12 chapter 10. Vector Algebra

1 P a g e Mathematics Notes for Class 12 chapter 10. Vector Algebra A vector has direction and magnitude both but scalar has only magnitude. Magnitude of a vector a is denoted by a or a. It is non-negative

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Tuesday, August 13, 2013 8:30 to 11:30 a.m., only.

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Tuesday, August 13, 2013 8:30 to 11:30 a.m., only Student Name: School Name: The possession or use of any communications

### 3. INNER PRODUCT SPACES

. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.

### Week 1 Chapter 1: Fundamentals of Geometry. Week 2 Chapter 1: Fundamentals of Geometry. Week 3 Chapter 1: Fundamentals of Geometry Chapter 1 Test

Thinkwell s Homeschool Geometry Course Lesson Plan: 34 weeks Welcome to Thinkwell s Homeschool Geometry! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson plan

### 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

### 15.062 Data Mining: Algorithms and Applications Matrix Math Review

.6 Data Mining: Algorithms and Applications Matrix Math Review The purpose of this document is to give a brief review of selected linear algebra concepts that will be useful for the course and to develop

### Cabri Geometry Application User Guide

Cabri Geometry Application User Guide Preview of Geometry... 2 Learning the Basics... 3 Managing File Operations... 12 Setting Application Preferences... 14 Selecting and Moving Objects... 17 Deleting

### 2010 Solutions. a + b. a + b 1. (a + b)2 + (b a) 2. (b2 + a 2 ) 2 (a 2 b 2 ) 2

00 Problem If a and b are nonzero real numbers such that a b, compute the value of the expression ( ) ( b a + a a + b b b a + b a ) ( + ) a b b a + b a +. b a a b Answer: 8. Solution: Let s simplify the

### REVIEW OF ANALYTIC GEOMETRY

REVIEW OF ANALYTIC GEOMETRY The points in a plane can be identified with ordered pairs of real numbers. We start b drawing two perpendicular coordinate lines that intersect at the origin O on each line.

### Geometry Unit 5: Circles Part 1 Chords, Secants, and Tangents

Geometry Unit 5: Circles Part 1 Chords, Secants, and Tangents Name Chords and Circles: A chord is a segment that joins two points of the circle. A diameter is a chord that contains the center of the circle.

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, August 13, 2015 8:30 to 11:30 a.m., only.

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, August 13, 2015 8:30 to 11:30 a.m., only Student Name: School Name: The possession or use of any communications

PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient

### 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

### Geometry 1. Unit 3: Perpendicular and Parallel Lines

Geometry 1 Unit 3: Perpendicular and Parallel Lines Geometry 1 Unit 3 3.1 Lines and Angles Lines and Angles Parallel Lines Parallel lines are lines that are coplanar and do not intersect. Some examples

### Section 9-1. Basic Terms: Tangents, Arcs and Chords Homework Pages 330-331: 1-18

Chapter 9 Circles Objectives A. Recognize and apply terms relating to circles. B. Properly use and interpret the symbols for the terms and concepts in this chapter. C. Appropriately apply the postulates,

### National 5 Mathematics Course Assessment Specification (C747 75)

National 5 Mathematics Course Assessment Specification (C747 75) Valid from August 013 First edition: April 01 Revised: June 013, version 1.1 This specification may be reproduced in whole or in part for

### 2.1. Inductive Reasoning EXAMPLE A

CONDENSED LESSON 2.1 Inductive Reasoning In this lesson you will Learn how inductive reasoning is used in science and mathematics Use inductive reasoning to make conjectures about sequences of numbers

### SAT Math Hard Practice Quiz. 5. How many integers between 10 and 500 begin and end in 3?

SAT Math Hard Practice Quiz Numbers and Operations 5. How many integers between 10 and 500 begin and end in 3? 1. A bag contains tomatoes that are either green or red. The ratio of green tomatoes to red

### The Area of a Triangle Using Its Semi-perimeter and the Radius of the In-circle: An Algebraic and Geometric Approach

The Area of a Triangle Using Its Semi-perimeter and the Radius of the In-circle: An Algebraic and Geometric Approach Lesson Summary: This lesson is for more advanced geometry students. In this lesson,

### Section 1: How will you be tested? This section will give you information about the different types of examination papers that are available.

REVISION CHECKLIST for IGCSE Mathematics 0580 A guide for students How to use this guide This guide describes what topics and skills you need to know for your IGCSE Mathematics examination. It will help

### FURTHER VECTORS (MEI)

Mathematics Revision Guides Further Vectors (MEI) (column notation) Page of MK HOME TUITION Mathematics Revision Guides Level: AS / A Level - MEI OCR MEI: C FURTHER VECTORS (MEI) Version : Date: -9-7 Mathematics

### Geometry Enduring Understandings Students will understand 1. that all circles are similar.

High School - Circles Essential Questions: 1. Why are geometry and geometric figures relevant and important? 2. How can geometric ideas be communicated using a variety of representations? ******(i.e maps,

### Unit 2 - Triangles. Equilateral Triangles

Equilateral Triangles Unit 2 - Triangles Equilateral Triangles Overview: Objective: In this activity participants discover properties of equilateral triangles using properties of symmetry. TExES Mathematics

### A vector is a directed line segment used to represent a vector quantity.

Chapters and 6 Introduction to Vectors A vector quantity has direction and magnitude. There are many examples of vector quantities in the natural world, such as force, velocity, and acceleration. A vector

### TIgeometry.com. Geometry. Angle Bisectors in a Triangle

Angle Bisectors in a Triangle ID: 8892 Time required 40 minutes Topic: Triangles and Their Centers Use inductive reasoning to postulate a relationship between an angle bisector and the arms of the angle.

### Path Tracking for a Miniature Robot

Path Tracking for a Miniature Robot By Martin Lundgren Excerpt from Master s thesis 003 Supervisor: Thomas Hellström Department of Computing Science Umeå University Sweden 1 Path Tracking Path tracking

### Lecture 4 DISCRETE SUBGROUPS OF THE ISOMETRY GROUP OF THE PLANE AND TILINGS

1 Lecture 4 DISCRETE SUBGROUPS OF THE ISOMETRY GROUP OF THE PLANE AND TILINGS This lecture, just as the previous one, deals with a classification of objects, the original interest in which was perhaps

### 2. If C is the midpoint of AB and B is the midpoint of AE, can you say that the measure of AC is 1/4 the measure of AE?

MATH 206 - Midterm Exam 2 Practice Exam Solutions 1. Show two rays in the same plane that intersect at more than one point. Rays AB and BA intersect at all points from A to B. 2. If C is the midpoint of

### Conjectures for Geometry for Math 70 By I. L. Tse

Conjectures for Geometry for Math 70 By I. L. Tse Chapter Conjectures 1. Linear Pair Conjecture: If two angles form a linear pair, then the measure of the angles add up to 180. Vertical Angle Conjecture:

### Algebra Geometry Glossary. 90 angle

lgebra Geometry Glossary 1) acute angle an angle less than 90 acute angle 90 angle 2) acute triangle a triangle where all angles are less than 90 3) adjacent angles angles that share a common leg Example:

### 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

### December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in two-dimensional space (1) 2x y = 3 describes a line in two-dimensional space The coefficients of x and y in the equation

### 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

### cos Newington College HSC Mathematics Ext 1 Trial Examination 2011 QUESTION ONE (12 Marks) (b) Find the exact value of if. 2 . 3

1 QUESTION ONE (12 Marks) Marks (a) Find tan x e 1 2 cos dx x (b) Find the exact value of if. 2 (c) Solve 5 3 2x 1. 3 (d) If are the roots of the equation 2 find the value of. (e) Use the substitution

### 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

### In mathematics, there are four attainment targets: using and applying mathematics; number and algebra; shape, space and measures, and handling data.

MATHEMATICS: THE LEVEL DESCRIPTIONS In mathematics, there are four attainment targets: using and applying mathematics; number and algebra; shape, space and measures, and handling data. Attainment target

### SOLID MECHANICS TUTORIAL MECHANISMS KINEMATICS - VELOCITY AND ACCELERATION DIAGRAMS

SOLID MECHANICS TUTORIAL MECHANISMS KINEMATICS - VELOCITY AND ACCELERATION DIAGRAMS This work covers elements of the syllabus for the Engineering Council exams C105 Mechanical and Structural Engineering

### 1 Solution of Homework

Math 3181 Dr. Franz Rothe February 4, 2011 Name: 1 Solution of Homework 10 Problem 1.1 (Common tangents of two circles). How many common tangents do two circles have. Informally draw all different cases,

### 12. Parallels. Then there exists a line through P parallel to l.

12. Parallels Given one rail of a railroad track, is there always a second rail whose (perpendicular) distance from the first rail is exactly the width across the tires of a train, so that the two rails

### GEOMETRY COMMON CORE STANDARDS

1st Nine Weeks Experiment with transformations in the plane G-CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point,