INTERPOLATION

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INTERPOLATION APPROXIMATION
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Curve algorithm

• General curve shape may be generated using method
  of
• Interpolation (also known as curve fitting)
• Curve will pass through control points
• Approximation
• Curve will pass near control points may
  interpolate the start and end points.

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Curve algorithm
interpolation
approximation
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Interpolation vs approximation
curve must pass through control points
curve is influenced by control points
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Parametric equation of line Vector equation of
a line
a
u

• P(t) a ut

• u (b-a)

• P(t) a (b-a)t

X(t) ax (bx ax)t Y(t) ay (by
ay)t Z(t) az (bz az )t
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Linear interpolation
B
t1
A
t0

• P(t) A(1-t) Bt
• In matrix form
• P(t) A B . .

X(t) Y(t) Z(t)
-1 1 1 0
t 1
in animation - path , morphing
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Interpolation Curves

• Curve is constrained to pass through all control
  points
• Given points P0, P1, ... Pn, find lowest degree
  polynomial which passes through the points x(t)
  an-1tn-1 .... a2t2 a1t a0 y(t)
  bn-1tn-1 .... b2t2 b1t b0

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Interpolating curve piecewise linear

• Curve defined by multiple segments (linear)
• Segments joints known as KNOTS
• Requires too many data points for most shape
• Representation not flexible enough to editing

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Interpolating curve piecewise polynomial

• Segments defined by polynomial functions
• Segments join at KNOTS
• Most common polynomial used is cubic (3rd order)
• Segment shape controlled by two or more adjacent
  control points.

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Knot points

• Location where segments join referred to as knots
• Knots may or may not coincide with control points
  in interpolating curves.

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Curve continuity

• Concern is continuity at knots.
• Continuity conditions
• Point continuity (no slope or curvature
  restriction / no gap)
• Tangent continuity (same slope at knot)
• Curvature continuity ( same slope and curvature
  at knot)

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Curve continuity

• Continuity - Cn
• C0 continuity continuity of endpoint only or
  continuity of position.
• C1 continuity is tangent continuity or first
  derivative of position
• C2 continuity is curvature continuity or second
  derivative of position.

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Curve continuity
C0
C1
C2
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Interpolation curves

• Typically possess curvature continuity
• Shape defined by
• Endpoint and control point location
• Tangent vectors at knots
• Curvature at knots

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Interpolation vs. Approximation Curves

• Interpolation Curve over constrained ? lots of
  (undesirable?) oscillations
• Approximation Curve more reasonable?

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Approximation techniques

• Developed to permit greater design flexibility in
  the generation of free form curves
• Common methods in modern CAD systems, bezier,
  b-spline, NURBS
• Employ control points (set of vertices that
  approximate the curve)

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Approximation techniques

• Curves do not pass directly through points
  (except start and end)
• Intermediate points affect shape as if exerting a
  pull on the curve.
• Allow user to set shape by pulling out curve
  using control point location.

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Example bezier curve
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Cubic Bézier Curve

• 4 control points
• Curve passes through first last control point
• Curve is tangent at P1 to (P1-P2) and at P4 to
  (P4-P3)