Geometric Modeling 91.580.201 Surfaces Mortenson Chapter 6 and Angel Chapter 9.

Geometric Modeling91.580.201

SurfacesMortenson Chapter 6and Angel Chapter 9

Surface Basics• Surface: Locus of a point moving with 2 degrees of freedom.• Some types of equations to describe curves:

– Intrinsic• No reliance on external frame of reference.• Lack of robustness of surface characteristics under repeated transformations.• Discussion deferred until we study more differential geometry.

– Explicit• Value of dependent variable in terms of independent variable(s)• e.g. z = f (x,y)• Lack of robustness of surface characteristics under repeated transformations.

– Implicit• e.g. f (x,y,z) = 0

– Parametric• Express value of each spatial variable in terms of independent variables (the

parameters)• e.g. for parameters u and w in 3D:

x = x (u,w)y = y (u,w)z = z (u,w)

source: Mortensonsource: Mortenson

Explicit Form• Value of dependent

variable in terms of independent variables– e.g. z = f (x,y)

• Axis-dependent• Can be hard to

represent a transformed and bounded surface.

• Sample surface-fitting procedure: determine aij coefficients from data points:

m

i

n

j

iiij yxayxz

0 0

),(


Implicit Form• General form: f (x,y,z) = 0

– f (x,y,z) is polynomial in x, y, z such that:

• Axis-dependent• Examples:

– Plane: Equation is linear in all its variables. – Quadric: Second-degree equation.

• Can represent using vectors, scalars and a type identifier.

– Right circular cylinder» One vector gives a point on its axis» One vector defines axis direction» Scalar gives radius

• Type testing requires robust floating-point computations.

kji

kjiijk zyxa

,,0


Implicit Form: Quadric Surfaces (continued)

• Type testing:


• Insert equations from p. 181


• Classification:


• Insert scanned Table 6.1 from p. 181


• Quadric Surfaces of Revolution:– Rotate conic curve about its axis– Canonical position:

• Center or vertex at origin• Axes of symmetry coincide with

coordinate axes.


• Insert scanned Table 6.2 from p. 183

Parametric Form

• Express value of each spatial variable in terms of independent variables (the parameters)– e.g. for parameters u, w in

3D:x = x (u,w)y = y (u,w)z = z (u,w)

• For a rectangular surface patch, typically

• Patches can be joined to

form composite parametric surfaces.

]1,0[, wu


Parametric Form (continued)

• Sample patch: rectangular segment of x, y plane

x = (c - a)u + a

y = (d - b)w + b

z = 0

• Here:– Curves of constant w

are horizontal lines.– Curves of constant u

are vertical lines.



• Parametric sphere of radius r, centered on (x0,y0,z0):

2,0 ,

2,

2 wheresinsincoscoscos 000

wuurzzwuryywurxx



• Parametric ellipsoid centered on (x0,y0,z0):

2,0 ,

2,

2 wheresinsincoscoscos 000

wuuczzwubyywuaxx



• Parametric surface of revolution:

2,0 ,]1,0[ where)(sin)(cos)( wuuzzwuxywuxx

partial view


4 Typical Types of Parametric Curves

• Interpolating– Curve passes through all control points.

• Hermite– Defined by its 2 endpoints and tangent vectors at endpoints.– Interpolates all its control points.– Not invariant under affine transformations.– Special case of Bezier and B-Spline.

• Bezier– Interpolates first and last control points.– Curve is tangent to first and last segments of control polygon.– Easy to subdivide.– Curve segment lies within convex hull of control polygon.– Variation-diminishing.– Special case of B-spline.

• B-Spline– Not guaranteed to interpolate control points.– Invariant under affine transformations.– Curve segment lies within convex hull of control polygon.– Variation-diminishing.– Greater local control than Bezier.

Control points influence curve shape.

source: Mortenson, source: Mortenson, AngelAngel

Interpolating

• Interpolates all control points.• Geometric form:

• Rarely used due to lack of derivative continuity at curve segment join points.

source: Angelsource: Angel

n

iii ubu

0

)()( pp

cubic case with equally spaced parameter values

Hermite

• Geometric form (cubic case):

• Hermite curves can provide C1 continuity at curve segment join points.

234

233

232

231

)(

2)(

32)(

132)(

uuuF

uuuuF

uuuF

uuuF


14031201 )()()()()( uu uFuFuFuFu ppppp

Bezier• Geometric form (cubic case):

• Bezier curves can provide C1 continuity at curve segment join points.

n

iini uBu

0, )()( pp


inini uu

i

nuB

)1()(,

Bernstein polynomials.

n+1 = number of control points = degree + 1

Adding a control point elevates degree by 1.

Convex combination, so Bezier curve points all lie within convex hull of control polygon.

n

ini uB

0, 1)(

Rational form is invariant under perspective transformation:where hi are projective space coordinates (weights)

n

inii

n

iinii

uBh

uBhu

0,

0,

)(

)()(

pp

B-Spline

n

iiKi uNu

0, )()( pp

otherwise 0)(

if 1)(

1,

11,

uN

tutuN

i

iii

• Geometric form (non-uniform, non-rational case), where K controls degree (K -1) of basis functions:

• Cubic B-splines can provide C2 continuity at curve segment join points.

Convex combination, so B-spline curve points all lie within convex hull of control polygon.

n

iKi uN

0, 1)(

Rational form (NURBS) is invariant under perspective transformation, where hi are projective space coordinates (weights).

n

iKii

n

iiKii

uNh

uNhu

0,

0,

)(

)()(

pp


1

1,1

1

1,,

)()()()()(

iki

kiki

iki

kiiki tt

uNut

tt

uNtuuN

ti are knot values that relate u to the control points.

Uniform case: space knots at equal intervals of u.

Repeated knots move curve closer to control points.

N

N

N N

N

N N

N N

Geometric Modeling 91.580.201 Surfaces Mortenson Chapter 6 and Angel Chapter 9.

Documents

mortenson slide

y plane x

y axisdependent

implicit form general

parametric form express

surfaces mortenson chapter

parameters u

y lack of robustness