Geometric ModelingGeometric Modelingfaculty.missouri.edu/.../lecture-notes/modeling... · • Geometric constructionGeometric construction – Specify a planar curve profile on y-z

Geometric ModelingGeometric Modeling

CECS461 Computer Graphics II University of Missouri at Columbia

OverviewOverview3D Shape Primitives:3D Shape Primitives:p• Points

– Vertices.

p• Points

– Vertices.Vertices.• Curves

– Lines polylines curves

Vertices.• Curves

– Lines polylines curvesLines, polylines, curves.• Surfaces

– Triangle meshes splines subdivision surfaces

Lines, polylines, curves.• Surfaces

– Triangle meshes splines subdivision surfacesTriangle meshes, splines, subdivision surfaces, implicit surfaces, particles.

• Solids

Triangle meshes, splines, subdivision surfaces, implicit surfaces, particles.

• Solids


So dsSo ds

Basic ShapesBasic Shapes


Fundamental ShapesFundamental Shapes






CurvesCurves• Lines.• Lines.Lines.• Polynomials.

L

Lines.• Polynomials.

L• Lagrange curves.• Hermite curves.• Lagrange curves.• Hermite curves.• Bezier curves.• B-Splines.• Bezier curves.• B-Splines.B Splines.• NURBS.

S bdi i i h

B Splines.• NURBS.

S bdi i i hCECS461 Computer Graphics II University of Missouri at Columbia

• Subdivision schemes.• Subdivision schemes.

Mathematical FormulationsMathematical Formulations


Linear interpolationLinear interpolation


SurfacesSurfaces• Planes• Planes• Triangle meshes.• Tensor-product surfaces.• Triangle meshes.• Tensor-product surfaces.Tensor product surfaces.

– Hermite surface, Bezier surface, B-spline surfaces, NURBS.

• No-tensor product surfaces.

Tensor product surfaces.– Hermite surface, Bezier surface, B-spline surfaces, NURBS.

• No-tensor product surfaces.p– Sweeping surface, ruling surface, etc.

• Subdivision surfaces.

p– Sweeping surface, ruling surface, etc.

• Subdivision surfaces.• Implicit surfaces.• Particle systems.• Implicit surfaces.• Particle systems.


Particle systems.Particle systems.

Parametric PolynomialsParametric Polynomials


PolynomialsPolynomials

• No intuitive insight.• No intuitive insight.• Difficult for piecewise smooth curves.• Difficult for piecewise smooth curves.


Cubic PolynomialsCubic Polynomials




How to define a curve?How to define a curve?• Specify a set of points for interpolation and/or • Specify a set of points for interpolation and/or p y p p

approximation for some ui

p y p papproximation for some ui

• Specify the derivatives at some locations for some ui• Specify the derivatives at some locations for some ui

• A combination of both.• A combination of both.


Lagrange CurveLagrange Curve• The curve interpolates all the control points.• The curve interpolates all the control points.The curve interpolates all the control points.• Unwanted oscillation.

The curve interpolates all the control points.• Unwanted oscillation.


Lagrange CurveLagrange Curveaa

)()()( ,

,

0,

,

uLaaa

uLaaa

uc nnyn

xnn

yn

xn

,, aa znzn

)(

)(,0

j

n

ijj

n

uu

uL

)()(

,0ji

n

ijj

i

uuuL


Kronecker delta functionKronecker-delta function

jiji

uL ijjni ,0

,1)(

ji,0


Cubic Hermite CurveCubic Hermite Curve




Cubic Hermite CurveCubic Hermite Curve• Constraints: two end-points and two tangents at• Constraints: two end-points and two tangents atConstraints: two end points and two tangents at

end-points.Constraints: two end points and two tangents at end-points.


Cubic Hermite CurveCubic Hermite Curve• Solve the linear equation:• Solve the linear equation:Solve the linear equation:Solve the linear equation:






Why Cubic CurvesWhy Cubic Curves• The lowest-degree polynomials to be able to interpolate two• The lowest-degree polynomials to be able to interpolate two

specified endpoints with specified derivatives at each endpoint.

Th l d h l i 3D

specified endpoints with specified derivatives at each endpoint.

Th l d h l i 3D• The lowest-degree curves that are nonplanar in 3D.• The lowest-degree curves that are nonplanar in 3D.

• Lower degree has too little flexibility.• Lower degree has too little flexibility.

• Higher-degree is unnecessarily complex, exhibit undesiredwiggles.

• Higher-degree is unnecessarily complex, exhibit undesiredwiggles.


Geometric RepresentationsGeometric Representations• Continuous representation• Continuous representationContinuous representation

– Explicit representation– Implicit representation

Continuous representation– Explicit representation– Implicit representation– Implicit representation

• Discrete representationT i l h

– Implicit representation• Discrete representation

T i l h– Triangle meshes– Points/Particles– Triangle meshes– Points/Particles


Deformable SurfaceDeformable Surface

Continuous Models Discrete Models

Implicit Representation Particle SystemsDiscrete MeshesExplicit

Representation

Subdivision Surface

Algebraic Surface

Level-Set

NURBS

Superquadric


B-Splines

SurfacesSurfaces• Triangle meshes.• Triangle meshes.g• Tensor-product surfaces.

– Hermite surface, Bezier surface, B-spline surfaces,

g• Tensor-product surfaces.

– Hermite surface, Bezier surface, B-spline surfaces,Hermite surface, Bezier surface, B spline surfaces, NURBS.

• Non-tensor product surfaces.

Hermite surface, Bezier surface, B spline surfaces, NURBS.

• Non-tensor product surfaces.p– Sweeping surface, ruling surface, etc.

• Subdivision surfaces.

p– Sweeping surface, ruling surface, etc.

• Subdivision surfaces.• Implicit surfaces.• Particle systems• Implicit surfaces.• Particle systemsCECS461 Computer Graphics II University of Missouri at Columbia

Particle systems.Particle systems.

Triangle MeshesTriangle Meshes


Quadratic SurfacesQuadratic Surfaces• Implicit representation• Implicit representationImplicit representation

=0S h

Implicit representation=0

S h• Sphere• Sphere

• Ellipsoid• Ellipsoid


Tensor Product SurfaceTensor Product Surface• From curves to surfaces• From curves to surfaces• A simple curve example (Bezier)• A simple curve example (Bezier)

where u [0,1]where u [0,1]where u [0,1]

• Consider pi is a curve pi(v)• In particular, if pi is also a bezier curve, where

where u [0,1]

• Consider pi is a curve pi(v)• In particular, if pi is also a bezier curve, whereIn particular, if pi is also a bezier curve, where

v [0,1]In particular, if pi is also a bezier curve, where v [0,1]


From curve to surfaceFrom curve to surface• Then we have• Then we haveThen we haveThen we have


Bezier SurfaceBezier Surface


B Splines SurfaceB-Splines Surface• B-Spline curves• B-Spline curves

T d t B liT d t B li

n

i kii uBpuc0 , )()(

• Tensor product B-splines• Tensor product B-splines

where u [0,1], and v [0,1]

f hi

where u [0,1], and v [0,1]

f hi• Can we get NURBS surface this way?• Can we get NURBS surface this way?


B Splines SurfaceB-Splines Surface


Tensor Product PropertiesTensor Product Properties• Inherit from their curve generators.• Inherit from their curve generators.Inherit from their curve generators.• Continuity across boundaries

I t l ti d i ti t l

Inherit from their curve generators.• Continuity across boundaries

I t l ti d i ti t l• Interpolation and approximation tools.• Interpolation and approximation tools.


NURBS SurfaceNURBS Surface


Surface of RevolutionSurface of Revolution


Surface of RevolutionSurface of Revolution• Geometric construction• Geometric constructionGeometric construction

– Specify a planar curve profile on y-z plane– Rotate this profile with respect to z-axis

Geometric construction– Specify a planar curve profile on y-z plane– Rotate this profile with respect to z-axis– Rotate this profile with respect to z-axis

• Procedure-based model– Rotate this profile with respect to z-axis

• Procedure-based model


Sweeping SurfaceSweeping Surface


Sweeping SurfaceSweeping Surface• Surface of revolution is a special case of a• Surface of revolution is a special case of aSurface of revolution is a special case of a

sweeping surface.• Idea: a profile curve and a trajectory curve

Surface of revolution is a special case of a sweeping surface.

• Idea: a profile curve and a trajectory curve• Idea: a profile curve and a trajectory curve.• Move a profile curve along a trajectory curve to

t i f

• Idea: a profile curve and a trajectory curve.• Move a profile curve along a trajectory curve to

t i fgenerate a sweeping surface.generate a sweeping surface.


Subdivision SchemeSubdivision SchemeOverview:Overview:Overview:

St t f i iti l t l l

Overview:

St t f i iti l t l l• Start from an initial control polygon.• Recursively refine it by some rules.• A smooth surface in the limit

• Start from an initial control polygon.• Recursively refine it by some rules.• A smooth surface in the limit• A smooth surface in the limit.• A smooth surface in the limit.

pxJpxs )(),(


Chaikin’s corner cutting scheme g








Catmull Clark SchemeCatmull-Clark Scheme


Catmull-Clark Scheme

Initial mesh Step 1


Step 2 Limit surface

Catmull-Clark SchemeCatmull Clark Scheme


Midedge schemeMidedge scheme


Midedge schemeMidedge scheme

(a) (b)

(d)(c)


(d)(c)

PointsPoints• Very popular primitives for modeling,• Very popular primitives for modeling,Very popular primitives for modeling,

animation, and rendering.Very popular primitives for modeling, animation, and rendering.


PointsPoints


Documents

Geometric ModelingGeometric Modelingfaculty.missouri.edu/.../lecture-notes/modeling... · • Geometric constructionGeometric construction – Specify a planar curve profile on y-z