Surfaces (Tensor product Surfaces) - University of Calgary

Advanced Geometric ModelingFaramarz Samavati

UNIVERSITY OF

CALGARY

Surfaces(Tensor product Surfaces)

Advanced Geometric Modeling

Faramarz Samavati

UNIVERSITY OF

CALGARY Parameter DomainA


Faramarz Samavati

UNIVERSITY OF

CALGARYExample of a simple parametric surface

How to compute?


Faramarz Samavati

UNIVERSITY OF

CALGARY Polygonalization

Choose a set of ui and vi (samples)

Compute the position Q(ui,vi) and normal n(ui,vi)

40 20


Faramarz Samavati

UNIVERSITY OF

CALGARY Generalization

Local parameterization


Faramarz Samavati

UNIVERSITY OF

CALGARY Parameter DomainA


Faramarz Samavati

UNIVERSITY OF

CALGARYDon’t forget still we havea parametric surfaces:

Free texture mapping


Faramarz Samavati

UNIVERSITY OF

CALGARY Use of u- and v-curves


Faramarz Samavati

UNIVERSITY OF

CALGARY Other directional curvesVenation schemes


Faramarz Samavati

UNIVERSITY OF

CALGARY


Faramarz Samavati

UNIVERSITY OF

CALGARY Embedded Curves

Other directions than u and v


Faramarz Samavati

UNIVERSITY OF

CALGARY General Blend!Blend “1” dimensional( instead of “0”dimensional objects) Linear Blend

P1

P0

P0 P1

Deformed Curve !!


Faramarz Samavati

UNIVERSITY OF

CALGARY Sketch Based Modeling

2D Sketch Input 3D Model


Faramarz Samavati

UNIVERSITY OF

CALGARY Artistic drawing with few strokes


Faramarz Samavati

UNIVERSITY OF

CALGARY

What is a default interpretation for the third dimension?

A round shapeBut it is not a surface of revolution!


Faramarz Samavati

UNIVERSITY OF

CALGARYSystem Overview

CreationPhase

EditingPhase

Parametric surfaces

optional


Faramarz Samavati

UNIVERSITY OF

CALGARYCreation Phase: modeling techniques?Constructive curves (strokes) to 3D surface

2D Strokes 3D Model


Faramarz Samavati

UNIVERSITY OF

CALGARY Use a cross section stroke


Faramarz Samavati

UNIVERSITY OF

CALGARY Rotational Blending SurfaceRotational for roundnessBlending for a continuous progression from

the first to the second stroke


Faramarz Samavati

UNIVERSITY OF

CALGARY Constructive curves


Faramarz Samavati

UNIVERSITY OF

CALGARY

v

What is algorithmic description of Rotational Blending Surface?

Parametric surface s(u,v) Circles form u-curvesThe blending curves are v-curves

u


Faramarz Samavati

UNIVERSITY OF

CALGARYUse the deformation of a ruled surface


Faramarz Samavati

UNIVERSITY OF

CALGARYCross Sectional Over-sketch

new cross section

+ =


Faramarz Samavati

UNIVERSITY OF

CALGARY Movies


Faramarz Samavati

UNIVERSITY OF

CALGARY

actual artusing our system

A complete example


Faramarz Samavati

UNIVERSITY OF

CALGARY Free Form Surfaces

Blend curves to obtain surfaces

Advanced Geometric ModelingFaramarz Samavati

UNIVERSITY OF

CALGARY


Faramarz Samavati

UNIVERSITY OF

CALGARY


Faramarz Samavati

UNIVERSITY OF

CALGARY


Faramarz Samavati

UNIVERSITY OF

CALGARY B-spline Surface B-spline surfaceRectangular surface patches, composite surface

We have 2 knot sequencesSurface, to be generated from Cartesian product of two

groups of curves


Faramarz Samavati

UNIVERSITY OF

CALGARY B-spline Surface


Faramarz Samavati

UNIVERSITY OF

CALGARY


Faramarz Samavati

UNIVERSITY OF

CALGARY


Faramarz Samavati

UNIVERSITY OF

CALGARY


Faramarz Samavati

UNIVERSITY OF

CALGARY Surface and Sums of B-splinesFirst component of surface

For a fixed u and v

Where

So, any component of surface needs several times evaluating of

General Mask


Faramarz Samavati

UNIVERSITY OF

CALGARY NURBSNURBS surfaces:

Efficient algorithm : use the algorithm “sum of B-Splines”


Faramarz Samavati

UNIVERSITY OF

CALGARY PropertiesCorner point interpolation for the standard knot sequenceAffine invarianceStrong convex hullLocal modificationNon-rational B-Spline and Bezier are special cases of NURBS

Surfaces (Tensor product Surfaces) - University of Calgary

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