16-01-2014, 04:59 PM
Curves and Surfaces
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Explicit Representation
Curve in 2D: y = f(x)
Curve in 3D: y = f(x), z = g(x)
Surface in 3D: z = f(x,y)
Problems:
– How about a vertical line x = c as y = f(x)?
– Circle y = f (r2 – x2)1/2 two or zero values for x
• Too dependent on coordinate system
• Rarely used in computer graphics
Constructive Solid Geometry (CSG)
Generate complex shapes with basic building
blocks
machine an object - saw parts off, drill holes
glue pieces together
This is sensible for objects that are actually made
that way (human-made, particularly machined
objects)
Algebraic Surfaces
• Special case of implicit representation
• f(x,y,z) is polynomial in x, y, z
• Quadrics: degree of polynomial w 2
• Render more efficiently than arbitrary surfaces
• Implicit form often used in computer graphics
• How do we represent curves implicitly?
Approximating Surfaces
• Use parametric polynomial surfaces
• Important concepts:
Join points for segments and patches
Control points to interpolate
Tangents and smoothness
Blending functions to describe interpolation
• First curves, then surfaces
Summary
Parametric Representations
Cubic Polynomial Forms
Hermite Curves
Bezier Curves and Surfaces