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COMP175: Computer Graphics

Lecture 14 Parametric Curves and Surfaces

Curves important for realism, aesthetic design, functional design

Can approximate with connected line segments

Piecewise linear functions

Can approximate with higher order functions

Piecewise line splines

Spline curve

A smooth curve specified in terms of a few control points

Interpolating curves

Some splines interpolate all of their control points

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Approximating curves

Others interpolate some control points and approximate others

Mechanical splines

Historically used in design and manufacturing

A lead spline weight or “duck” Using weights and a wooden spline to draw a spline curve

Mathematical curve representation

Desired spline qualities Interpolating vs. approximating curves Continuity and ‘smoothness’

Efficient rendering and processing Locating points on the curve Subdividing curves into smaller segments Measuring accuracy of fit

Representations of shape Explicit function Implicit function Parametric representation

Curve quality

Desirable properties: Continuity Interpolation Local control

Continuity

Determined by continuity at points where the piecewise polynomial curves are joined

Each polynomial curve is smooth along its length

Want to avoid abrupt changes at segment joints

Nested degrees of continuity: C-1, C0, C1, … , CN

C-1 continuity

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C0 continuity

Adjacent segments join at a common endpoint

C1 continuity

The first derivative is continuous across joints

(CN continuity: The nth derivative is continuous across joints)

Curve quality

Computer graphics often requires C1 continuity

Automotive industry often requires C2

Aircraft and race cars may require C4 or C5

Constraining curves

Constraints: interpolated points, tangent vectors, continuity

The coefficients of a polynomial curve are chosen to satisfy the specified constraints

Polynomial curve forms Hermite curves Bezier curves General spline curves: B-splines, non-uniform B-splines (NURBS), β-splines

Bezier Curves

Constraints for Bezier curves

One more control point than the order of the curve

Interpolate the control points specifying the endpoints of the curve

Approximate the other control points

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Constraints for Bezier curves

€

P(t) = Bi (t)Pii= 0

N

∑

Constraints for Bezier curves

Blending functions

Force the curve to interpolate the endpoints: At t = 0, B0(t) = 1 and all other Bi(t) = 0 At t = 1, BN(t) = 1 and all other Bi(t) = 0

The other two constraints control how much the curve “wiggles” between the endpoints

€

P(t) = Bi (t)Pii= 0

N

∑

Bernstein polynomials

One choice for the blending functions

where

Example, N = 1: line segment B0 = (1-t) B1 = t

€

Bi(t) = witi(1− t)N− i

€

wi =N!

(i!(N − i)!)

Bernstein polynomials for cubic Bezier curves

The Bernstein polynomials are all positive in [0,1]

P0 has the greatest influence at t = 0

P3 has the greatest influence at t = 1

Other control points have their greatest influence where their curves are maximum

Properties of Bezier curves

The Bezier blending functions sum to 1 for all t

Thus, all points P(t) on the Bezier curve must lie inside the convex hull defined by the curve’s control points

P0

P1

P2

P3

Quadratic Bezier curve

P1

P2 P0

Control points

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Quadratic Bezier curve

P1

P2 P0

Endpoint tangent direction vectors

Quadratic Bezier curve

P1

P2 P0

Convex hull

Quadratic Bezier curve

P1

P2 P0

Curve

Cubic Bezier curve

P2

P3 P0

Control points

P1

Cubic Bezier curve

Endpoint tangent directions

P2

P3 P0

P1

Cubic Bezier curve

Convex hull

P2

P3 P0

P1

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Cubic Bezier curve

Curve

P2

P3 P0

P1

More cubic Bezier curves

P2

P3

P0

P1

P1

P3 P0

P2

Rendering Bezier curves‏

Use the parametric form of the curve Approximate the curve by a set of line segments Evaluate P(t) at N values of ti, with the ti evenly spaced between 0 and 1 to obtain

Qi = P(ti)

More about displaying curves next time…

Q0

Q1

Q2

Q3 Q4 Q5

Q6

Q7

Q8

Bezier Curves: Summary

Constrained by a set of control points, {Pi} One more control point than the order of the curve

Interpolate the endpoints of the curve and approximate the other control points of the curve

Determine points on the curve using a set of blending functions

Used Bernstein polynomials

€

P(t) = Bi (t)Pii= 0

N

∑

€

Bi(t) = witi(1− t)N− i

€

wi =N!

(i!(N − i)!)

Curved Surfaces

Piecewise patches

Represent complex curved surfaces with piecewise continuous curved patches

Constrain patches to ensure surface continuity

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Linear interpolation – curves Fits the ‘simplest’ curve between two

points (i.e., a straight line)

P1

P2

P(t) = (1-t) P0 + t P1

Bi-linear interpolation – surfaces Fits the ‘simplest’ surface between four points

P10

P11

P00

P01

Linear interpolation – curves Fits the ‘simplest’ curve between two

points (i.e., a straight line)

P1

P2

P(t) = (1-t) P0 + t P1

P10

P11

P00

P01

P0(v) = (1-v)P00 + vP01

P1(v) = (1-v)P10 + vP11

v increasing

v increasing

Bi-linear interpolation – surfaces Fits the ‘simplest’ surface between four points

P11

P00

P01

Q0(u) = (1-u)P00 + uP10

Q1(u) = (1-u)P01 + uP11

u increasing

u increasing

Linear interpolation – curves Fits the ‘simplest’ curve between two

points (i.e., a straight line)

P1

P2

P(t) = (1-t) P0 + t P1

Bi-linear interpolation – surfaces Fits the ‘simplest’ surface between four points

Linear interpolation – curves Fits the ‘simplest’ curve between two

points (i.e., a straight line)

P1

P2

P(t) = (1-t) P0 + t P1

P10

P11

P00

P01

P(u,v) = (1-u)(1-v) P00 + (1-u)v P01 + u(1-v) P10 + uv P11

u increasing

v increasing

Bi-linear interpolation – surfaces Fits the ‘simplest’ surface between four points

Bi-linear surface patch

Forms a parametric expression for the curved surface

For a constant u = c, P(c,v) is a straight line For a constant v = b, P(u,b) is a straight line

… but the combined surface is curved

P10

P11

P00

P01

P(u,v) = (1-u)(1-v) P00 + (1-u)v P01 + u(1-v) P10 + uv P11

u increasing

v increasing

Bi-linear surface patch

Each line at a constant u or v is called an iso-parametric curve

For a bi-linear surface, each iso-parametric curve is a straight line P(0,v) is the line from P00 to P01

P(1,v) is the line from P10 to P11

P(u,0) is the line from P00 to P10

P(u,1) is the line from P01 to P11

P10

P11

P00

P01

P(u,v) = (1-u)(1-v) P00 + (1-u)v P01 + u(1-v) P10 + uv P11

u increasing

v increasing

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Bi-linear surface patch

Alternative to evaluating P(u, v) directly:

Compute Q0(v) = P(0,v) = (1-v)P00 + vP01

and Q1(v) = P(1,v) = (1-v)P10 + vP11

Then, P(u,v) = (1-u)Q0(v) + uQ1(v)

P10

P11

P00

P01

P(u,v) = (1-u)(1-v) P00 + (1-u)v P01 + u(1-v) P10 + uv P11

u increasing

v increasing

Q0(v)���

Q1(v)���

Two cubic Bezier curves can be used as the end curves of a ruled surface patch

P0(u) ���

P1(u) ���

Two cubic Bezier curves can be used as the end curves of a ruled surface patch

P0(u) ���

P1(u) ���

Ruled surface patch

P(u,v) = (1-v)P0(u) + vP1(u)

P0(u) = P(u,0) ���

P1(u) = P(u,1) ���

P(u,v3)

P(u,v2)

P(u,v1)

Swept surfaces

Bi-linear surface A surface swept out as we move a straight

line through space Restrict endpoints of the line to lie on the

two end lines

Ruled surface A surface swept out as we move a straight

line through space Restrict endpoints of the line to lie on the

two end curves

Swept surfaces

A tensor product surface A surface swept out as we move a curve through space Change the shape of the curve as it moves through space in some controlled

manner

A tensor product Bezier surface Sweeps a Bezier curve through space by moving control points of the Bezier curve

along a Bezier curve

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Tensor product Bezier patch

P0

P1

P2 P3

Tensor product Bezier patch

P0

P1

P2 P3

P0(v)���

P1(v)���

P2(v)���

P3(v)���

Tensor product Bezier patch

P0

P1

P2 P3

P0(v)���

P1(v)���

P2(v)���

P3(v)���

The original curve:

Control point curves:

€

P(u) = BiM (u)Pi

i= 0

M

∑

€

Pi(v) = B jN (v)Pij

j= 0

N

∑

P00

P10

P20 P30

P31

P32

P33

P21 P22

P23 P11 P12

P13 P01

P02

P03

Tensor product Bezier patch

€

P(u,v) =j= 0

N

∑ BiM B j

N (v)Piji= 0

M

∑

Tensor product Bezier patch

(Quadratic) curves followed by the control points

Original control points (cubic end curve)

Grid of control points

Surface

Controlling the surface

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Properties of Bezier patches

Affine invariance

Convex hull

Patch boundaries Four boundary curves given by

where u = 0, u = 1, v = 0, and v = 1

€

P(u,v) =j= 0

N

∑ BiM B j

N (v)Piji= 0

M

∑

Rendering Bezier patches

Determine a set of points, P(u,v), on the surface at equally spaced increments of u and v over [0,1]

v increasing

u increasing

Rendering Bezier patches

Determine a set of points, P(u,v), on the surface at equally spaced increments of u and v over [0,1]

When points close, the surface between them is approx. planar

Approximate each patch by a quadrilateral and render these

v increasing

u increasing

Bezier patch continuity

To model more complex shapes, use a mesh of piecewise polynomial Bezier surface patches

When joining patches, need continuity constraints

Bezier patch continuity: C0

Joined patches are touching all along their shared edge

The control polygons of the two patches are the same along the shared edge

Bezier patch continuity: C1

Joined edges have collinear tangent vectors across their shared edge

A severe constraint:

Joining a cubic patch to one existing patch 8 of the new patch’s control points are constrained

Joining a cubic patch to two existing patchs even more of the new patch’s control points are constrained