Lecture 17 parametric curves and surfacesbpbettig/MAE455/Lecture_17...Parametric Representation of...

MAE 455 Computer-Aided Design and Drafting

Parametric Representation of Curves and Surfaces

How does the computer compute curves and surfaces?

MAE 455 Computer-Aided Design and Drafting 2

Types of Curve Equations

• Implicit Form

• Explicit Form

• Parametric Form

CAD uses primarily the parametric form.

0 ,222==+ zRyx

0 ,)( 22=−±= zxRxy

0)( ,sin)( ,cos)( === θθθθθ zRyRx


Straight Line & Conic Curves

• Straight Line:

• Circle:

• Ellipse:

• Hyperbola:

• Parabola:

x(u) = R cosu

y(u) = R sinu z(u) = 0

x(u) = a cosu

y(u) = b sinu z(u) = 0

x(u) = a coshu

y(u) = b sinhu z(u) = 0

x(u) = c u2

y(u) = u z(u) = 0

Line segments and conic arcs are established by specifying ranges for u

(e.g., 0 ≤ u ≤ π/2; or 0 ≤ u ≤ 1).

Transformation equations are used to rotate and translate the curves to the desired

orientation and location.

x(u) = x0 + dx u

y(u) = y0 + dy u z(u) = z0 + dz u


Polynomial Freeform Curves

• Freeform curves (and even straight lines and arcs) are

represented in CAD using polynomials. E.g.:

=

)(

)(

)(

)(

uz

uy

ux

uP

3

3

2

210 uuu aaaa +++=

+++

=

3

0,3

2

0,20,10,0 uauauaa3

1,3

2

1,21,11,0 uauauaa +++3

2,3

2

2,22,12,0 uauauaa +++

+

=

2,0

1,0

0,0

a

a

a

+

2

2,2

1,2

0,2

u

a

a

a

+

u

a

a

a

2,1

1,1

0,1

3

2,3

1,3

0,3

u

a

a

a

( )10 ≤≤ u


B-Spline Curve

• The coefficients a0, a1, a2, a3 are hard for a designer to specify because the geometric affect is not intuitive.

• CAD software therefore uses “B-Spline” curves.• B-Spline curves are controlled using “control points” .

B-Spline curve with:

– Degree: 3

– Num. control points: 4

P0 (u = 0)

P3(u = 1)

P1 P2

“control points”

(“poles” in NX)

“control” polyline


B-Spline Curve Equation

• The B-spline curve equation is:

• Note that at each point of the curve each control point Pi

has an influence given by Ni,k(u).

• n is the num. control points – 1

• k is the degree + 1

• t are a series of increasing

numbers (“knots”).



Degree: 7

Num. control points: 8

• Using a smaller degree limits the influence of each control point.

Degree: 3

Num. control points: 8

N0,4

N1,4 N2,4N3,4 N4,4 N5,4 N6,4

N7,4

u

N0,8

N3,8

N6,8N5,8N1,8 N4,8

N2,8

N7,8

u

Blue triangles represent knots

MAE 455 Computer-Aided Design and Drafting

u = 0

u = tlast

u = 0

u = tlast

8


Degree: 7; Num. control points: 8

Evaluating

this point

• Note that the B-spline curve is composed of n - k + 2 segments, each of degree k-1. Here the segments are shown separated by the pink circles (which also represent knot locations).

• Making the degree smaller brings the curve closer to the

control points.

Degree: 3; Num. control points: 8

The colored lines

show the influence

of the control points.


Closed versus Open Curves

• A B-Spline curve can be “open” or “closed”

“open” “closed”


B-Spline Curve Properties

• Open curves always pass through 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 curve will result if the control points are created in the reverse order (only u = 0 will be at the reverse end).

• The curve is always inside the convex hull of the control polygon:

Figure is from: K. Lee, “Principles of CAD/CAM/CAE Systems,” Addison-Wesley, 1999


B-Spline Curve Properties

• Also, if the degree is too high, moving a control

point at the beginning of the curve will result in

changes to the curve at the other end.

Second order interpolation

Eleventh order interpolation

• Be careful with using too high a degree. Higher order curves are inherently more wavy.


NURBS curves

• NURBS means “Non-uniform Rational B-Spline”.

• NURBS have a weighting factor hi associated with each

control point.

• In NURBS curves the knot values do not have to be

uniformly spaced.

• NURBS curves are useful because they allow exact

representation of conic curves.

h0 = 1 h2 = 1

h1 = 0.707


Types of Surface Equations

• Non-parametric - explicit

• Parametric

• Non-parametric – implicit

– e.g. sphere:2222

Rzyx =++

222),( zxRzxy −−±=

=

=

)sin(

)cos()sin(

)cos()cos(

),(

),(

),(

),(

vR

vuR

vuR

vuz

vuy

vux

vuP


Primitive Surfaces

• Plane P(u, v) = u i + v j + 0 k

x

y

z

u

v

• Cylinder P(u, v) = R cos u i + R sin u j + v k

x

y

z

u

v

R


Primitive Surfaces

• Plane

• Cylinder

• Sphere

• Cone

• Torus

P(u, v) = u i + v j + 0 k

P(u, v) = R cos u i + R sin u j + v k

P(u, v) = R cos u cos v i + R sin u cos v j+ R sin v k

P(u, v) = m v cos u i + m v sin u j + v k

P(u, v) = (R + r cos v) cos u i+ (R + r cos v) sin u j + r sin v k

Transformation equations are used to rotate and translate

these surfaces into the desired orientation and location.


The B-Spline Surface

• The B-Spline surface is an extension of the B-Spline curve concept to one higher dimension.

• It uses a grid of control points, evaluated in uand v to surface points.


B-Spline Surface

Properties:

• Boundaries are B-Spline curves.

• Intuitive control of surface interior.

• Derivatives (surface normals) can be evaluated

using same algorithm used to evaluate points.

• Surface is inside convex hull of control points

• NURBS surfaces can exactly represent rounded

surfaces (e.g., cylindrical and spherical surfaces).


Extrude Operation

1. Start with NURBS curve:

4. Duplicate the weightings in each row. jjj hhh == ,1,0

2. Duplicate the control points.jj PP =,0

3. Create another duplicate row of control points translated by da.

aPP djj +=,1

Lecture 17 parametric curves and surfacesbpbettig/MAE455/Lecture_17...Parametric Representation of...

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