INTERPOLATION & APPROXIMATION

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.

Curve algorithm

interpolation approximation

Interpolation vs approximation

x

f(x)

x

f(x)

x

f(x)

x

f(x)

curve must pass through control points

curve is influenced by control points

Parametric equation of line = Vector equation of a line

• P(t) = a + ut

a

u

P

u 2u

P

tu

P

a

u

b

•P(t) = a + (b-a)t

•u = (b-a)

t=0t=1

0<=t<=1

P

t=0.25

P

t=0.5

P

t=0.75

X(t) = ax + (bx – ax)tY(t) = ay + (by – ay)tZ(t) = az + (bz – az )t

Linear interpolation

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

A

B

t=0

t=1

X(t)Y(t)Z(t)

-1 11 0

t1

in animation : - path , morphing

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

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

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.

Knot points

• Location where segments join referred to as knots

• Knots may or may not coincide with control points in interpolating curves.

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)

• 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.

Curve continuity

Curve continuity

C0 C1

C2

Interpolation curves

• Typically possess curvature continuity

• Shape defined by– Endpoint and control point location– Tangent vectors at knots– Curvature at knots

Interpolation vs. Approximation Curves

• Interpolation Curve – over constrained → lots of (undesirable?) oscillations

• Approximation Curve – more reasonable?

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)

• 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.

Approximation techniques

Example – bezier curve

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)