Parametric Representation of Curves and Surfaces

Transcription

1 Parametric Representation of Crves and Srfaces How does the compter compte crves and srfaces? MAE 455 Compter-Aided Design and Drafting

2 Types of Crve Eqations Implicit Form Explicit Form Parametric Form x y( x) + y = R, z = 0 = ± R x, z = 0 x( θ ) = R cosθ, y( θ ) = R sinθ, z( θ ) = 0 CAD ses primarily the parametric form. MAE 455 Compter-Aided Design and Drafting

3 Straight Line & Conic Crves Straight Line: Circle: Ellipse: Hyperbola: Parabola: x() = x 0 + d x y() = y 0 + d y z() = z 0 + d z x() = R cos y() = R sin z() = 0 x() = a cos y() = b sin z() = 0 x() = a cosh y() = b sinh z() = 0 x() = c y() = z() = 0 Line segments and conic arcs are established by specifying ranges for (e.g., 0 π/; or 0 1). Transformation eqations are sed to rotate and translate the crves to the desired MAE 455 Compter-Aided Design and Drafting 3 orientation and location.

4 Polynomial Freeform Crves Freeform crves (and even straight lines and arcs) are represented in CAD sing polynomials. E.g.: P( ) = x( ) y( ) z( ) a a1 + a a3 MAE 455 Compter-Aided Design and Drafting 4 = = = a + a + a + a a a 0,0 1,0,0 a3,0 3 0,1 + a1,1 + a,1 a3,1 0, + a1, + a, a3, a + a + 0,0 0,1 0, + a a a 1,0 1,1 1, a + a a,0,1, 3 3 a + a a 3,0 3,1 3, ( 0 1) 3

5 B-Spline Crve The coefficients a 0, a 1, a, a 3 are hard for a designer to specify becase the geometric affect is not intitive. CAD software therefore ses B-Spline crves. B-Spline crves are controlled sing control points. B-Spline crve with: Degree: 3 Nm. control points: 4 P 1 P control polyline P 3 ( = 1) P 0 ( = 0) control points ( poles in NX) MAE 455 Compter-Aided Design and Drafting 5

6 B-Spline Crve Eqation The B-spline crve eqation is: n is the nm. control points 1 k is the degree + 1 t are a series of increasing nmbers ( knots ). Note that at each point of the crve each control point P i has an inflence given by N i,k (). MAE 455 Compter-Aided Design and Drafting 6

7 B-Spline Crve Eqation Using a smaller degree limits the inflence of each control point. N 0,8 N 7,8 Degree: 7 Nm. control points: 8 N 1,8 N N N 6,8,8 3,8 N 4,8 N 5,8 Degree: 3 Nm. control points: 8 N 0,4 N 7,4 N 1,4 N N 3,4 N 4,4,4 N 5,4 N 6,4 Ble triangles represent knots MAE 455 Compter-Aided Design and Drafting 7

8 B-Spline Crve Eqation Making the degree smaller brings the crve closer to the control points. Evalating this point The colored lines show the inflence of the control points. = 0 = t last Degree: 7; Nm. control points: 8 = 0 = t last Degree: 3; Nm. control points: 8 Note that the B-spline crve is composed of n - k + segments, each of degree k-1. Here the segments are shown separated by the pink MAE circles 455 Compter-Aided (which also Design represent and Drafting knot locations). 8

9 Closed verss Open Crves A B-Spline crve can be open or closed open closed MAE 455 Compter-Aided Design and Drafting 9

10 B-Spline Crve Properties Open crves always pass throgh the first and last point. The tangent at first point is given by the direction from the first control point to the second. The tangent at last point is given by the direction from the second last control point to the last. The same crve will reslt if the control points are created in the reverse order (only = 0 will be at the reverse end). The crve is always inside the convex hll of the control polygon: MAE 455 Compter-Aided Design and Drafting 10 Figre is from: K. Lee, Principles of CAD/CAM/CAE Systems, Addison-Wesley, 1999

11 B-Spline Crve Properties Be carefl with sing too high a degree. Higher order crves are inherently more wavy. Second order interpolation Eleventh order interpolation Also, if the degree is too high, moving a control point at the beginning of the crve will reslt in changes to the crve at the other end. MAE 455 Compter-Aided Design and Drafting 11

12 NURBS crves NURBS means Non-niform Rational B-Spline. NURBS have a weighting factor h i associated with each control point. In NURBS crves the knot vales do not have to be niformly spaced. NURBS crves are sefl becase they allow exact representation of conic crves. h 1 = MAE 455 Compter-Aided Design and Drafting 1 h 0 = 1 h = 1

13 MAE 455 Compter-Aided Design and Drafting 13 Types of Srface Eqations Non-parametric - explicit Parametric Non-parametric implicit e.g. sphere: R z y x = + + ), ( z x R z x y = ± = = ) sin( ) )cos( sin( ) )cos( cos( ), ( ), ( ), ( ), ( v R v R v R v z v y v x v P

14 Primitive Srfaces Plane P(, v) = i + v j + 0 k z z v y x R v y x Cylinder P(, v) = R cos i + R sin j + v k MAE 455 Compter-Aided Design and Drafting 14

15 Primitive Srfaces Plane Cylinder Sphere Cone Tors P(, v) = i + v j + 0 k P(, v) = R cos i + R sin j + v k P(, v) = R cos cos v i + R sin cos v j + R sin v k P(, v) = m v cos i + m v sin j + v k P(, v) = (R + r cos v) cos i + (R + r cos v) sin j + r sin v k Transformation eqations are sed to rotate and translate these srfaces into the desired orientation and location. MAE 455 Compter-Aided Design and Drafting 15

16 The B-Spline Srface The B-Spline srface is an extension of the B- Spline crve concept to one higher dimension. It ses a grid of control points, evalated in and v to srface points. MAE 455 Compter-Aided Design and Drafting 16

17 B-Spline Srface Properties: Bondaries are B-Spline crves. Intitive control of srface interior. Derivatives (srface normals) can be evalated sing same algorithm sed to evalate points. Srface is inside convex hll of control points NURBS srfaces can exactly represent ronded srfaces (e.g., cylindrical and spherical srfaces). MAE 455 Compter-Aided Design and Drafting 17

18 Extrde Operation 1. Start with NURBS crve:. Dplicate the control points. 3. Create another dplicate row of control points translated by da. 4. Dplicate the weightings in each row. MAE 455 Compter-Aided Design and Drafting 18 P 1, h h = 0, j P = P 0, j j = P + j j = 1, j da h j