CS 552 Computer Graphics Lecture 25 Surface Representation

Recap • Bi-Cubic Surface representation o u- edge o v- edge o Combination with

Objective • After completing this lecture, students will be able to o Derive the

Sca�olding Construction • The sca�olding construction (or de Casteljau algorithm) • Extension of the

Subdividing Rectangular Patches • To change the surface shape, if there are not enough

Example • Starting with the 2× 3 control points • Elevation: 1× 2 to

Thank you Next Lecture: Surface Representation

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CS 552: Computer Graphics Lecture 25: Surface Representation

Recap • Bi-Cubic Surface representation o u- edge o v- edge o Combination with correction • Subdividing bi-cubic surface o Representing with a new set of control points o Limits can be specified

Objective • After completing this lecture, students will be able to o Derive the mathematical expressions for Ø Bezier surface Ø B-spline surface o Construct Bezier surface • Calculate expressions for o Surface subdivision o Degree elevation

Bézier Curve • Bézier equation

Bezier Surface •

Example •

Sca�olding Construction • The sca�olding construction (or de Casteljau algorithm) • Extension of the idea discussed in Bezier curves

Subdividing Rectangular Patches • To change the surface shape, if there are not enough points • Just adding points is not a good solution because o this changes the shape of the surface • A better solution is to subdivide the patch into four connected surface patches Step 1: Select values for u and w Step 2: Apply de Casteljau Algo (column wise) Step 3: Apply de Casteljau Algo (row wise)

Degree Elevation •

Degree Elevation for Surface •

Example • Starting with the 2× 3 control points • Elevation: 1× 2 to 2× 3

B-Spline Surface •

Thank you Next Lecture: Surface Representation