Curve and surface (đồ họa máy TÍNH SLIDE)

Control Points• We can specify control points to draw the curve... Polynomial Curves• Could make a curve that passes through n control points out of a degree n-1 polynomial – Regression,

Curve and Surface

• Subdivision Curves & Surfaces

• Appendix: Subdivision Masks

How to describe this curve?

Control Points

• We can specify control points to draw the curve

Control Points

• Control points are not a unique specification

• A polynomial is a function of the form:

f(t) = ant n + an-1t n-1 + … + a1t + a0

• n is the degree of the polynomial

• The order of the polynomial is the number of

Polynomial Curves

• Could make a curve that passes through n

control points out of a degree (n-1) polynomial

– Regression, Lagrange interpolation, etc.

Polynomial Curves

• Polynomial curves wiggle too much when forced

to fit more and more points

• In technical terms this is called overfitting

Why Parametric Curves?

• Parametric curves are very flexible

• They are not required to be functions

• Curves can be multi-valued with respect to any coordinate system

Particle Motion

• A parametric curve P(t) describes the motion of

an imaginary particle through space at time t

• We can compute the velocity of the particle:

• The tangent line at P(t0) to the curve is:

Parametric Polynomial Curves

• A parametric polynomial curve is a parametric curve

• where each function x(t), y(t)

n

i i i

Curve Drawing

• We want:

– Predictable control: Curves don’t wiggle

– Multiple values: Curves of arbitrary length

– Local control: Local edits have local effects

– Versatility: Be able to draw any curve

– Continuity: Smoothness guarantees

• Spline curvesgive us all of these

Spline Curves

• The word splinecomes from ship building with wood

• A wooden plank is forced between fixed posts,

called “ducks”

• Real-world splines are still being used for designing ship hulls, automobiles, and aircraft fuselage and

wings

www.abm.org

• We have seen parametric polynomials Let’s

look at the other terms:

– Piecewise

– Continuity conditions

– Advantages: Provides flexibility

– Problem: How do we fit the pieces together?

Parametric Continuity

• C0: Curves are joined

– “watertight” curve / mesh

• C1: First derivative is continuous

– d/dt Q(t) = velocity is the same.

– “looks smooth, no facets”

• C2: Second derivative is continuous

– d 2 /dt 2 Q(t) = acceleration is

the same (important for animation

and shading)

Geometric Continuity

• G0: Curves are joined

• G1: First derivatives are proportional at the joint point

– The tangent vectors have the same directions, but not

necessarily the same magnitude

– Velocity of a moving point is not continuous

• G2: First and second derivatives are proportional at joint point

– Acceleration of the point is not continuous

• Parametric continuity of order n implies geometric

continuity of order n, but not vice-versa.

Specifying Splines

• Control Points - a set of

points that influence the

curve's shape

• Hull - the lines that

connect the control points

Parametric Cubic Curves

• In order to assure C2 continuity our functions

must be of at least degree three

• Here's what a 2D parametric cubic function looks like:

• In matrix form:

Parametric Cubic Curves

• This is a cubic function in 3D:

• To avoid the dependency on the dimension we will use the following notation:

Parametric Cubic Curves

• What does the derivative of a cubic curve look like?

Solving for Coefficients

• Problem: Polynomial coefficients (the c’s) are

amazingly bad control knobs

• Usually, we want to control the curve in terms of what

it does — passing through control points, etc.

• The whole story of polynomial splines is deriving

their coefficients given a set of control points and

continuity conditions

• Approach:

– State what we want the curve to do

– Solve for coefficients that satisfy the constraints set by the

• Subdivision Curves & Surfaces

• Appendix: Subdivision Masks

Cubic Hermite Specification

• Given:

– Two control points (P1, P2).

– Tangents (derivatives) at the knot points (P’1, P’2):

Hermite Spline

• 4 segments, (P1, P2) and (P’1, P’2) for each

segment

Building it up…

• Cubic curve equations:

• Boundary constraints:

Solve for the c’s