SurfacesSurfaces

Chiew-Lan Tai

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Surfaces

Reading

Required Hills Section 11.11 Hearn & Baker, sections 8.11, 8.13

Recommended Sections 2.1.4, 3.4-3.5, 3D Computer

Graphics, Watt

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Surfaces

Mathematical surface representations Explicit z = f(x,y) (a.k.a. “height field”) What if the surface isn’t a function?

Implicit g(x,y,z) = 0 x2 + y2 + z2 = 1

Parametric S(u,v) = (x(u,v), y(u,v), z(u,v)) x(u,v) = r cos 2v sinu y(u,v) = r sin 2v sinu z(u,v) = r cos u 0≤u≤1, 0≤v≤1

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Surfaces

Tensor product Bézier surfaces

Given a grid of control point Vij, forming a control net, construct a surface S(u,v) by:

Each row in u-direction is control points of a curve Vi(u).

V0(u), …, Vn(u) at a specific u are the control points of a curve parameterized by v (i.e. S(u,.v))

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Surfaces

Tensor product Bezier surfaces, cont. Let’s walk through the steps:

Which points are interpolated by the surface? V0(u) and V3(u)

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Surfaces

Tensor product Bezier surfaces, cont. Writing it out explicitly: Linear Bezier surface:

Quadratic Bezier surface?

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Surfaces

Matrix form

For the case of cubic Bezier surface:

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Surfaces

Tensor product B-spline surfaces

Like spline curves, we can piece together a sequence of Bezier surfaces to make a spline surface. If we enforce C2 continuity and local control, we get B-spline curves:

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Surfaces

Tensor product B-spline surfaces

Use each row of B control points in u to generate Bezier control points in u. Treat each row of Bezier control points in v direction as B-spline control points to

generate Bezier control points in v direction.

u

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Surfaces

Tangent plane and surface normal

S

S

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Surfaces

Partial Derivatives

0

1

2

3

]1[),(

2

23 u

u

MVMvvvvuSu

TBezierBezier

1

]0123[),(2

3

2

u

u

u

MVMvvvuSv

TBezierBezier

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Surfaces

Surface normal

S

S S

S

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Surfaces

Other construction methods

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Surfaces

Rotational surfaces (surface of revolution) Rotate a 2D profile curve about an axis

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Surfaces

Profile curve, C(u)

Constructing surfaces of revolution Given a curve C(u) in the yz-plane:

Let Ry() be a rotation about the y-axis Find: A surface S(u,v) obtained by applying Ry() on C(u)

S(u,v) = Ry(v) [0, Cy(u), Cz(u), 1]t

y

xz

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Surfaces

Surface of revolution

S(u,v) = ( Cz(u) sin(v), Cy(u), Cz(u) cos(v))

y

z

x

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Surfaces

General sweep surface

Given a planar profile curve C(u) and a general space curve T(v) as the trajectory, sweep C(u) along the trajectory

How to orient C(u) as it moves along T(v)?

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Surfaces

Orientating C(u) Define a local coordinate frame at any point along the trajectory The Frenet frame (t,n,b) is the natural choice

As we move along T(v), the Frenet frame (t,b,n) varies smoothly (inflection points where curvature goes to zero needs special treatment)

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Surfaces

Sweep Surfaces

Orient the profile curve C(u) using the Frenet frame of the trajectory T(v): Put C(u) in the normal plane. Place Oc at T(v). Align xc of C(u) with b. Align yc of C(u) with n.

Sweep surface

S(u,v) = T(v) + n(v)Cx(u) + b(v) Cy(u)

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Surfaces

Variations

Several variations are possible: Scale C(u) as it moves, possibly using length of T(v) as a

scale factor. Morph C(u) into some other curve C’(u) as it moves along

T(v)

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Surfaces

Summary

What to take home: How to construct tensor product Bezier surfaces How to construct tensor product B-spline surfaces How to construct surfaces of revolution How to construct sweep surfaces from a profile and a

trajectory curve with a Frenet frame

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Surfaces

Subdivision SurfacesWhat’s wrong with B-spline/NURBS surfaces?

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Surfaces

Subdivision curves

Idea: repeatedly refine the control polygon

curve is the limit of an infinite process.

i

i

PC

PPP

lim

210

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Surfaces

Subdivision curves

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Surfaces

Chaikin’s algorithm In 1974, Chaikin introduced the following “corner-

cutting” scheme: Start with a piecewise linear curve Repeat

Insert new vertices at the midpoints (the splitting step)

Average two neighboring vertices (the average step)

Averaging mask(0.5, 0.5)

With this averaging mask, in the limit, the resulting curve is a quadratic B-spline curve.

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Surfaces

Local subdivision mask Subdivision mask: ¼ (1, 2, 1) Splitting and averaging:

Applying the step recursively, each vertex converges to a specific point

With this mask, in the limit, the resulting curve is a cubic B-spline curve.

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Surfaces

General subdivision process

After each split-average step, we are closer to the limit surface.

Can we push a vertex to its limit position without infinite subdivision? Yes!

We can determine the final position of a vertex by applying the evaluation mask. The evaluation mask for cubic B-spline is 1/6 (1,4, 1)

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Surfaces

DLG interpolating scheme (1987)

Algorithm: splitting step introduces midpoints, and averaging step

only changes these midpoints Dyn-Levin-Gregory scheme:

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Surfaces

Building Complex Models

This simple idea can be extended to build subdivision surfaces.

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Surfaces

Subdivision surfaces Iteratively refine a control polyhedron (or control mesh) to

produce the limit surface using splitting and averaging step:

There are two types of splitting schemes: vertex schemes face schemes

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Surfaces

Vertex schemes

A vertex surrounded by n faces is split into n subvertices, one for each face:

Doo-Sabin subdivision:

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Surfaces

Face schemes Each quadrilateral face is split into four subfaces:

Catmull-Clark subdivision:

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Surfaces

Face scheme, cont.

Each triangular face is split into four subfaces:

Loop subdivision:

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Surfaces

Averaging step Averaging masks:

nn

QQQnQ n

)(

)( 1

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Surfaces

Adding creases Sometimes, a particular feature such as a crease should be

preserved. we just modify the subdivision mask.

This gives rise to G0 continuous surfaces.

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Surfaces

Creases

Here’s an example using Catmull-Clark surfaces: