Representing Curves and
Surfaces
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Introduction
• We need smooth curves and surfaces in
many applications:
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model real world objects
computer-aided design (CAD)
high quality fonts
data plots
artists sketches
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Introduction
• Most common representation for surfaces:
– polygon mesh
– parametric surfaces
– quadric surfaces
• Solid modeling
– don’t miss the next episode...
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Introduction
• Polygon mesh:
– set of connected planar surfaces bounded by
polygons
– good for boxes, cabinets, building exteriors
– bad for curved surfaces
– errors can be made arbitrarily small at the cost
of space and execution time
– enlarged images show geometric aliasing
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Introduction
• Parametric polynomial curves:
– point on 3D curve = (x(t), y(t), z(t))
– x(t), y(t), and z(t) are polynomials
– usually cubic: cubic curves
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Introduction
• Parametric bivariate (two-variable)
polynomial surface patches:
– point on 3D surface = (x(u,v), y(u,v), z(u,v))
– boundaries of the patches are parametric
polynomial curves
– many fewer parametric patches than
polynomial patches are needed to approximate
a curved surface to a given accuracy
– more complex algorithms though
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Parametric cubic curves
• Polylines and polygons:
– large amounts of data to achieve good accuracy
– interactive manipulation of the data is tedious
• Higher-order curves:
– more compact (use less storage)
– easier to manipulate interactively
• Possible representations of curves:
– explicit, implicit, and parametric
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Parametric cubic curves
• Explicit functions:
– y = f(x), z = g(x)
– impossible to get multiple values for a single x
• break curves like circles and ellipses into segments
– not invariant with rotation
• rotation might require further segment breaking
– problem with curves with vertical tangents
• infinite slope is difficult to represent
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Parametric cubic curves
• Implicit equations:
– f(x,y,z) = 0
– equation may have more solutions than we
want
• circle: x² + y² = 1, half circle: ?
– problem to join curve segments together
• difficult to determine if their tangent directions
agree at their joint point
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Parametric cubic curves
• Parametric representation:
– x = x(t), y = y(t), z = z(t)
– overcomes problems with explicit and implicit
forms
– no geometric slopes (which may be infinite)
– parametric tangent vectors instead (never
infinite)
– a curve is approximated by a piecewise
polynomial curve
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Parametric cubic curves
• Why cubic?
– lower-degree polynomials give too little
flexibility in controlling the shape of the curve
– higher-degree polynomials can introduce
unwanted wiggles and require more
computation
– lowest degree that allows specification of
endpoints and their derivatives
– lowest degree that is not planar in 3D
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Parametric cubic curves
• Kinds of continuity:
– G0: two curve segments join together
– G1: directions of tangents are equal at the joint
– C1: directions and magnitudes of tangents are
equal at the joint
– Cn: directions and magnitudes of n-th
derivative are equal at the joint
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Parametric cubic curves
• Major types of curves:
– Hermit
• defined by two endpoints and two tangent vectors
– Bezier
• defined by two endpoints and two other points that
control the endpoint tangent vectors
– Splines
• several kinds, each defined by four points
• uniform B-splines, non-uniform B-splines, ß-splines
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Parametric cubic curves
• General form:
x(t )  a x t 3  bx t 2  c x t  d x
y (t )  a y t 3  by t 2  c y t  d y
z (t )  a z t 3  bz t 2  c z t  d z
ax a y az 
b b b 
x
y
z

C
T  [t 3 t 2 t 1]
 cx c y cz 


d
d
d
 x
y
z

Q(t )  [ x(t ) y (t ) z (t )]  T  C  T  M  G
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Parametric cubic curves
• It is not necessary to choose a single
representation, since it is possible to convert
between them.
• Interactive editors provide several choices,
but internally they usually use NURBS,
which is the most general.
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Parametric bicubic surfaces
• Generalization of parametric cubic curves.
Q( s, t )  T  C (t )  T  M  G (t )
• For each value of s there is a family of
curves in t.
• Major kinds of surfaces:
– Hermit, Bezier, B-spline
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Parametric bicubic surfaces
• Displaying bicubic surfaces:
– brute-force iterative evaluation is very
expensive (the surface is evaluated 20,000
times if step in parameters is 0.01)
– forward-difference methods are better, but still
expensive
– fastest is adaptive subdivision, but it might
create cracks
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Quadric surfaces
• Implicit form:
f ( x, y, z )  ax 2  by 2  cz 2  2dxy  2eyz  2 fxz  2 gx  2hy  2 jz  k  0
• Particularly useful for molecular modeling.
• Alternative to rational surfaces if only
quadric surfaces are being represented.
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Quadric surfaces
• Reasons to use them:
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easy to compute normal
easy to test point inclusion
easy to compute z given x and y
easy to compute intersections of one surface
with another
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Summary
• Polygon meshes
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well suited for representing flat-faced objects
seldom satisfactory for curved-faced objects
space inefficient
simpler algorithms
hardware support
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Summary
• Piecewise cubic curves and bicubic surfaces
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permit multiple values for a single x or y
represent infinite slopes
easier to manipulate interactively
can either interpolate or approximate
space efficient
more complex algorithms
little hardware support
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