CS-378: Game Technology Lecture #5: Curves and Surfaces Prof. Okan Arikan University of Texas, Austin Thanks to James O’Brien, Steve Chenney, Zoran Popovic,

CS-378: Game Technology

Lecture #5: Curves and Surfaces

Prof. Okan ArikanUniversity of Texas, Austin

Thanks to James O’Brien, Steve Chenney, Zoran Popovic, Jessica Hodgins

V2005-14-1.1

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Administrative

Final ProjectFirst deliverable

Homework is out

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Today

General curve and surface representations

Splines and other polynomial bases

Points

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Geometry Representations

Constructive Solid Geometry (CSG)Parametric

Polygons

Subdivision surfaces

Implicit SurfacesPoint-based Surface

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Object made by CSGConverted to polygons

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Object made by CSGConverted to polygonsConverted to implicit surface

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CSG on implicit surfaces

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Point-based surface descriptions

Ohtake, et al., SIGGRAPH 2003

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Subdivision surface(different levels of refinement)

Images from Subdivision.org

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Various strengths and weaknessesEase of use for design

Ease/speed for rendering

Simplicity

Smoothness

Collision detection

Flexibility (in more than one sense)

Suitability for simulation

many others...

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Parametric Representations

Curves:

Surfaces:

Volumes:

and so on...

Note: a vector function is really n scalar functions

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Same curve/surface may have multiple formulae

Parametric Rep. Non-unique

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Simple Differential Geometry

Tangent to curve

Tangents to surface

Normal of surface

Also: curvature, curve normals, curve bi-normal, others...Degeneracies: or

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Discretization

Arbitrary curves have an uncountable number of parameters

i.e. specify function value at all points on real number line

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Discretization

Arbitrary curves have an uncountable number of parameters

Pick complete set of basis functions

Polynomials, Fourier series, etc.

Truncate set at some reasonable point

Function represented by the vector (list) of

The may themselves be vectors

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Polynomial Basis

Power Basis

The elements of are linearly independant

i.e. no good approximation

Skipping something would lead to bad results... odd stiffness

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Specifying a Curve

Given desired values (constraints) how do we determine the coefficients for cubic power basis?

For now, assume

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Specifying a Curve


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Specifying a Curve


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Specifying a Curve


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Specifying a Curve


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Specifying a Curve


Hermite basis functions

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Specifying a Curve


Hermite basis functions

Probably not to scale.

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Hermite Basis

Specify curve by Endpoint values

Endpoint tangents (derivatives)

Parameter interval is arbitrary (most times)

Don’t need to recompute basis functions

These are cubic HermiteCould do construction for any odd degree

derivatives at end points

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Cubic Bézier

Similar to Hermite, but specify tangents indirectly

Note: all the control points are points in space, no tangents.

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Cubic Bézier

Similar to Hermite, but specify tangents indirectly

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Cubic Bézier

Plot of Bézier basis functions

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Changing Bases

Power basis, Hermite, and Bézier all are still just cubic polynomials

The three bases all span the same space

Like different axes in

Changing basis

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Useful Properties of a Basis

Convex HullAll points on curve inside convex hull of control points

Bézier basis has convex hull property

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Invariance under class of transforms

Transforming curve is same as transforming control points

Bézier basis invariant for affine transforms

Bézier basis NOT invariant for perspective transforms

NURBS are though...

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Local supportChanging one control point has limited impact on entire curve

Nice subdivision rules

Fast evaluation scheme

Interpolation -vs- approximation

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DeCasteljau Evaluation

A geometric evaluation scheme for Bézier

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Joining

If you change a, b, or c you must change the others

But if you change a, b, or c you do not have to change beyond those three. *Local Support*

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Tensor-Product Surfaces

Surface is a curve swept through space

Replace control points of curve with other curves

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Hermite Surface Bases

Plus symmetries...

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Hermite Surface Hump Functions

Plus symmetries...

Points

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A Thought ExperimentLaser scanners Millions to billions of points

Typical imageAt most a few million pixels

More points than pixels...

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“Point-Based Graphics”

Surfaces represented only by pointsMaybe normals also

No topology

How can we doRendering

Modeling opperations

Simulation

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Rendering

For each point draw a little “splat” Use associated normal for shading

Possibly apply texture

If “splats” are small compared to spacing then gaps result


Splatting too many points wouldwaste time

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Rendering

“QSplat” algorithm Build hierarchical tree of the pointsUse bounding spheres to estimate size of clustersRender clusters based on screen sizeUse cluster-normals for internal nodes

From Rusinkiewicz and Levoy, SIGGRAPH 2000.

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Rendering


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Rendering


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Rendering


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Rendering


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Defining a Surface

Two related methodsSurface is a point attractor

Point-set surfaces

Implicit surface

Multi-level Partition of Unity Implicits

Implicit Moving Least-Squares

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Point-Set Surfaces

Surface is the attractor of a repeated projection processFind nearby points

Fit plane (weighted)

Project into plane

Repeat

Does it converge?How to weight points?

From Amenta and Kil, SIGGRAPH 2004.

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Implicit Moving Lest-Squares

Define a scalar function that is zero passing through all the points

From Shen, et al., SIGGRAPH, 2004.

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Implicit Moving Lest-Squares

From Shen, et al., SIGGRAPH, 2004.

Function is zero on boundaryDecreases in outward direction

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Moving Least-Square Interpolation

Standard Least SquareStandard Least Square

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Moving Least SquareMoving Least Square

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Least Square Moving Least Square

InterpolatingInterpolating

ApproximatingApproximating

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Editing Operations

Implicit function can beCombined w/ booleans

Warped

Offset

Composed

And more...


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Editing Operations


Warped

Offset

Composed

And more...


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Warped

Offset

Composed

And more...

Editing Operations


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Point-Based Simulation

MLS originated in mechanics literature

Natural use in graphics for animation

From Mueller, et al., SCA, 2004.

CS-378: Game Technology Lecture #5: Curves and Surfaces Prof. Okan Arikan University of Texas, Austin Thanks to James O’Brien, Steve Chenney, Zoran Popovic,

Documents

implicit surface slide

implicit surfaces slide

cubic power basis

polynomial basis power

nonunique slide

odd stiffness slide

geometry representations

desired values constraints