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Subdivision Curves & Surfaces and Fractal Mountains.PowerPoint Presentation

Subdivision Curves & Surfaces and Fractal Mountains.

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### Subdivision Curves & Surfaces and Fractal Mountains.

Outline

Outline

CS184 – Spring 2011

Outline

- Review Bézier Curves
- Subdivision Curves
- Subdivision Surfaces
- Quad mesh (Catmull-Clark scheme)
- Triangle mesh (Loop scheme)

- Fractal Mountains

Review of Bézier CurvesDeCastlejau Algorithm

001

011

Ignore funny notation at vertices!

(= CS 284 stuff )

Insert at t = ¾

111

000

Review of Bézier CurvesDeCastlejau Algorithm

001

011

0¾1

0¾¾

00¾

¾¾¾

¾¾1

¾11

Insert at t = ¾

111

000

Review of Bézier CurvesDeCastlejau Algorithm

001

011

0¾1

0¾¾

00¾

¾¾¾

¾¾1

¾11

Curve position and tangent for t = ¾

Insert at t = ¾

111

000

Subdivision of Bézier Curves

001

011

0¾1

0¾¾

00¾

¾¾¾

¾¾1

¾11

This also yields all control points for subdivision into 2 Bezier curves

Insert at t = ¾

111

000

Subdivision of Bézier Curves

001

011

0¾1

0¾¾

00¾

¾¾¾

¾¾1

¾11

Convex Hull Property!

Insert at t = ¾

111

000

Bézier Curves Summary

- DeCastlejau algorithm is good for
- Evaluating position(t) and tangent(t),
- Subdividing the curve into 2 subcurves with their own control polygons.

- Subdivision of Bézier curves and their convex hull property allows for:
- Adaptive rendering based on a flatness criterion,
- Adaptive collision detection using line segment tests.

Outline

- Review Bézier Curves
- Subdivision Curves
- Subdivision Surfaces
- Quad mesh (Catmull-Clark scheme)
- Triangle mesh (Loop scheme)

- Fractal Mountains

V20

Subdivision CurvesV30

An approximating scheme

Limit curve

V40

V10

- Subdivision is a recursive 2 step process
- Topological split
- Local averaging / smoothing

E20

V20

Subdivision CurvesV30

E10

E30

V40

V10

E40

- Subdivision is a repeated 2 step process
- Topological split
- Local averaging / smoothing

E21

V20

Subdivision CurvesV30

V21

V31

E11

E31

V41

V11

V40

V10

E41

- Subdivision is a repeated 2 step process
- Topological split
- Local averaging / smoothing

E21

V20

Subdivision CurvesV30

V21

V31

E11

E31

V41

V11

V40

V10

E41

- Subdivision is a repeated 2 step process
- Topological split
- Local averaging / smoothing

E21

V20

Subdivision CurvesV30

V21

V31

E11

E31

V41

V11

V40

V10

E41

- Subdivision is a repeated 2 step process
- Topological split
- Local averaging / smoothing

Subdivision Curve Summary

- Subdivsion is a recursive 2 step process:
- Topological split at midpoints,
- Local averaging/smoothing operator applied.

- Doubles the number of vertices at each step
- Subdivision curves are nothing new:
- Suitable averaging rules can yield uniform B-spline curves.

Outline

- Review Bézier Curves
- Subdivision Curves
- Subdivision Surfaces
- Quad mesh (Catmull-Clark scheme)
- Triangle mesh (Loop scheme)

- Fractal Mountains

Subdivision Overview

Control Mesh

Topological Split

Averaging

Limit Surface

- Subdivision is a two part process:
- Topological split
- Local averaging / smoothing

Subdivision Overview

Control Mesh

Generation 1

Generation 2

Generation 3

Repeated uniform subdivisions of the control mesh converge to the limit surface.

Limit surface can be calculated in closed form for stationary schemes(averaging mask does not change).

Outline

- Review Bézier Curves
- Subdivision Curves
- Subdivision Surfaces:
- Quad mesh (Catmull-Clark scheme)
- Triangle mesh (Loop scheme)

- Fractal Mountains

B-spline Surfaces

- A cubic B-spline surface patch is controlled by a regular 4x4 grid of control points

B-spline Surfaces

- 2 adjacent patches share 12 control points and meet with C2 continuity

B-spline Surfaces

- Requires a regular rectangular control mesh grid and all valence-4 vertices to guarantee continuity.

Catmull-Clark Subdivision Surface

- Yields smooth surfaces over arbitrary topology control meshes.
- Closed control mesh closed limit surface.
- Quad mesh generalization of B-splines
- C1 at non-valence-4 vertices,
- C2 everywhere else (B-splines).

- Also: Sharp corners can be tagged
- Allows for smooth and sharp features,
- Allows for non-closed meshes.

Catmull-Clark Subdivision

Gen 0

Gen 1

Gen 2

- Extraordinary vertices are generated by non-valence-4vertices and faces in the input mesh.
- No additional extraordinary vertices are created after the first generation of subdivision.

C20

V20

C10

C30

V30

F20

E20

F10

F30

E30

E40

V00

E10

V10

V40

En0

F40

Fn0

C40

Cn0

Vn0

Catmull-Clark Averaging(simple averaging)

n = valence

- Review Bézier Curves
- Subdivision Curves
- Subdivision Surfaces
- Quad mesh (Catmull-Clark scheme)
- Triangle mesh (Loop scheme)

- Fractal Mountains

Summary

- Subdivision is a 2 step recursive process:
- Topological split,
- Local averaging / smoothing.

- It is an easy way to make smooth objects
- of irregular shape
- of topologies other than rectangular (torus).

- Review Bézier Curves
- Subdivision Curves
- Subdivision Surfaces
- Quad mesh (Catmull-Clark scheme)
- Triangle mesh (Loop scheme)

- Fractal Mountains

Linear Fractal Mountains

Gen 0:

- 2-step recursive process:
- Subdivide chain by creating edge midpoints,
- Randomly perturb midpoint positions(proportional to subdivided edge length).

Gen 1:

Gen 2:

Gen 3:

Fractal Mountain Surfaces

Gen 0

Gen 1

Gen 2

- 2-step recursive process:
- Subdivide triangles at edge midpoints,
- Randomly perturb midpoint positions.

Fractal Mountains Summary

- 2-step recursive process:
- Topological split at edge midpoints,
- Random perturbation of midpoint positions.

- Triangle topological split maintains a water-tight connected mesh.
- Useful to make uneven, “natural” terrain.
- Often a low-order subdivision is good enough to control terrain-following vehicles.

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