Curves and Surfaces (cont’)
Amy Zhang
Conversion between Representations
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Example: Convert a curve from a cubic B-spline curve to
the Bézier form:
From Curve to Surface
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Recall that a parametrically defined curve in 3D is given by
three univariate functions:
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where 0 ≤u≤1
By varying u from 0 to 1, a curve is swept out in 3D
Similarly, a parametric surface is defined by three bivariate
functions:
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where 0 ≤u≤1 and 0 ≤v≤1
By varying u from 0 to 1while keeping v constant, a curve is
swept out in 3D
An infinity of such curves exists as v is varied from 0 to 1, and
this defines a surface in 3D
Parametric Surfaces
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A single surface element, also known as a patch, is the
surface traced out as the parameters (u,v) take all values
between 0 and 1
Complex surfaces are modelled using nets of individual
patches analogous to the way in which complex curves
are made up of individual curve segments
The teapot is made up of
32 Bézier patches
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Like in the case of curves, cubic basis functions, which are
a good compromise between complexity and flexibility,
are most commonly used in compute graphics
Bicubic parametric patches are defined over a rectangular
domain in u,v-space and the boundary curves of the patch
are themselves cubic polynomial curves
Parametric Surfaces
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Advantages of patch representation over polygon mesh
representation:
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Conciseness: A parametric representation of a surface is
“exact” and “economic”, and mass properties, like surface area
and volume, are extractable from such a description
Parametric Surfaces
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Deformation and shape change: Deformation of a
parametric surface can be achieved by moving the
control points that define it. The deformations will
appear just as smooth and accurately represented as
the un-deformed counterpart when rendered
Bicubic Surfaces
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A bicubic parametric patch can be expressed in terms of the
basis function and the control points:
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where pij is a set of 16 control points:
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The net of control points forms a polyhedron in cartesian
space and the position of the points in this space control
shape of the surface
Bicubic Surfaces
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Example: The effect of lifting one of the control point of a
Bézier patch
Bicubic Surfaces
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Matrix formulation:
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where
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The matrix M is the characteristic matrix of the particular
form e.g.,MB for Bézier patch and MBS for B-spline patch
The matrix P is a 4×4 matrix that contains the 16 control
points
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Bézier Patch
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A Bézier patch is defined in matrix notion by:
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and alternatively by:
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Note that the boundary curves of a Bézier patch are
themselves Bézier curves, e.g.,
The Bézier patch has
analogous properties to and
advantages of the Bézier curve
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Bézier Patch
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Differentiation:
Bézier Patch
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Differentiation at the corners:
Bézier Patch
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Joining 2 Bézier patches Q
and R with control points qij
and rij
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Zero order continuity: q3i= r0i
for i =0,…,3
First order continuity:(q3i–q2i)
=k(r1i–r0i)for i =0,…,3
Any corner point cannot
be moved without
controlling 8 adjacent
 The above constraints may be
points to maintain
too severe:
continuity
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Bézier Patch
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Recall that a Bézier curve can be rendered efficiently
by recursive subdivision
Bézier Patch
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Similarly, a Bézier Patch can be rendered by
successively subdividing the control polygons into a
series of polygons that represent the Bézier surface
to a given tolerance in curvature.
The process of recursive subdivision of a Bézier Patch
is called patch splitting.
The result polygons from patch splitting can be
passed to a conventional polygon render and
standard rendering techniques can be used to render
surfaces made up of nets of patches.
Bézier Patch
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Due to the orthogonality of the parametric directions
u& v, a Bézier patch can be split in one parameter
without considering the other
Bézier Patch
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Consider a patch to be made up of 4 curves, the
control points of which correspond to the rows /
columns of P
Splitting is then applied to these curves separately,
yielding 2 sets of 4 curves qij& rij
These 2 sets of 4 curves then give 2 subpatches of the
original patch
Bézier Patch
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A given patch is subdivided down until the convex hull is
deem to be sufficiently planar and the edges are
sufficiently linear:
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Flatness is tested by forming a plane between 3
corner control points and measuring how far the
remaining control points deviate from this plane
Linearity of the edges is tested by measuring how
far the interior control points of the edge deviate
from the line joining the end control points
Bézier Patch
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The patch is then turned into 2 triangles by taking the 4
corners of the patch as the triangle vertices
The surface normal at each corner is then computed as
the cross product of the 2 tangent vectors at the corner,
e.g.,
Bézier Patch
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The cracking problem will appear due to the independent
subdivision of adjacent subpatches:
Uniform B-spline Patch
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A uniform B-splinepatch is defined in matrix notion by:
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A composite B-spline surface is C2 continuous across the
patches
The B-spline patch has analogous properties to and
advantages of the B-spline curve
Converting a B-spline patch to a Bézier patch for
rendering:
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