# Curves and surfaces by il4Fo7Y

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```									Curves and surfaces

Splines
Two main principles:
• Many polygons to approximate curved
surfaces
• Subdivide surfaces into subsurfaces,
patches (curved)
With polygons
• increasing number of polygons
• decreasing sizes of the polygons
• can be tens of thousands of polygons
(cp. saxophone)
• much storage
With patches
• boundaries are mathematically defined
curves
• more advanced algorithms are required
• more processing/patch
• fewer patches
• no patch representation possible for
very complex objects (cp. saxophone)
Different curve approaches
General curves are usually defined from some
type of control points (methods given in
increasing complexity order):
• Connect points with straight lines
• Interpolation polynomial
• Least square approximation
• Splines, approximating (usual) or
interpolating
– Bézier splines
– B-splines
Interpolation vs approximation
Interpolation; curve
goes through the
control points

Approximation; curve
close to (or through)
the control points
Example spline curve
Example spline surface
A surface is a direct
extension of curves,
actually a set of
orthogonal curves
Curve properties
•   Control points (knots)
•   Multiple valued
•   Axis independance
•   Global or local control
•   Variation-diminishing
•   Versatility
•   Order of continuity, C0, C1, C2
Continuity
Function representation
Parametric (or vector-valued) functions in
3D:
• P(u) = (x(u) y(u) z(u)), curve
• P(u,v) = (x(u,v) y(u,v) z(u,v)), surface

Ex: P(u)=(cos u, sin u), 0≤u≤2(unit
circle)
Spline curves
Piecewise continuous polynomial sections
Interpolating or approximating (or a
combination)
A control graph (control polygon) connects the
control points
Different representations:
• set of boundary conditions
• matrix
• set of blending (weight) functions
Spline curve approach
Usually, a third degree polynomial in each
section
x(u)=axu3 + bxu2 + cxu + dx
y(u)=ayu3 + byu2 + cyu + dy
z(u)=azu3 + bzu2 + czu + dz
Plotting: (x(u),y(u),z(u)) is then written
using u:=0 step u until n, u small
Cubic Bézier Curves
Assume C1 (1st order continuity) and four
control points (knots) in each section
(polynomial), pk=(xk,yk,zk), k=0,1,2,3
a) Boundary conditions: (cp. f(x)=x3 =>
f’(x)=3x2
x(0) = x0
x(1) = x3
x’(0) = 3(x1 - x0)
x’(1) = 3(x3 - x2)
Similar for y and z
Cubic spline,cont’d
b) Matrix
Cubic spline,cont’d
c) Blending functions (Bernstein polynomials)
x(u) = x0.BEZ0,3(u) + … + x3.BEZ3,3(u)
där
Example circle as a spline
Example spline curves

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