march 1, 2009dr. muhammed al-mulhem1 ics 415 computer graphics hermite splines dr. muhammed...

March 1, 2009 Dr. Muhammed Al-Mulhem 1

ICS 415Computer Graphics

Hermite Splines

Dr. Muhammed Al-MulhemDr. Muhammed Al-Mulhem

March 1, 2009March 1, 2009

Dr. Muhammed Al-MulhemDr. Muhammed Al-Mulhem

March 1, 2009March 1, 2009


Spline

• A spline is a a flexible strip used to produce a A spline is a a flexible strip used to produce a smooth curve through a set of points.smooth curve through a set of points.

• A spline is a a flexible strip used to produce a A spline is a a flexible strip used to produce a smooth curve through a set of points.smooth curve through a set of points.


Representations of Curves

• A curve can be represented using:

– A sequence of points.

– A piecewise polynomial functions.

• A spline curve refers to any composite curve formed with polynomial sections.

• A curve can be represented using:

– A sequence of points.

– A piecewise polynomial functions.

• A spline curve refers to any composite curve formed with polynomial sections.


Specifying Curves

• A spline curve is specified as follows:

– Giving a set of coordinate positions, called control Points, which indicate the general shape of the curve.

– These points are then fitted with piecewise polynomial functions in one of two ways:

Interpolation

Approximation

• A spline curve is specified as follows:

– Giving a set of coordinate positions, called control Points, which indicate the general shape of the curve.

– These points are then fitted with piecewise polynomial functions in one of two ways:

Interpolation

Approximation


Specifying Curves

• Interpolation:

– When polynomial sections are fitted so that all the control points are on the curve, the resulting curve is said to interpolate the set of control points.

• Interpolation:

– When polynomial sections are fitted so that all the control points are on the curve, the resulting curve is said to interpolate the set of control points.


Specifying Curves

• Approximation

– When polynomial sections are fitted so that some or all the control points are not on the curve , the resulting curve is said to approximate the set of control points.

• Approximation

– When polynomial sections are fitted so that some or all the control points are not on the curve , the resulting curve is said to approximate the set of control points.


Parametric Continuity

• To ensure a smooth transition from one section of a piecewise parametric spline to the next, we impose various continuity conditions at the connection points.

• If each section of a spline curve is described with a set of parametric coordinate function of the form

x = x(u), y= y(u), z = z(u), u1 <= u <= u2

• We set parametric continuity by matching the parametric derivatives of adjoining curve sections at their common boundary.

• To ensure a smooth transition from one section of a piecewise parametric spline to the next, we impose various continuity conditions at the connection points.

• If each section of a spline curve is described with a set of parametric coordinate function of the form

x = x(u), y= y(u), z = z(u), u1 <= u <= u2

• We set parametric continuity by matching the parametric derivatives of adjoining curve sections at their common boundary.



• Various continuity conditions:

• C0 continuity (Zero-order parametric continuity):

– Means simply that curves meet. That is, the values of x, y, and z evaluated at u2 for the first curve section are

equal, respectively, to the values of x, y, and z evaluated at u1 of the next curve section.

• Various continuity conditions:

• C0 continuity (Zero-order parametric continuity):

– Means simply that curves meet. That is, the values of x, y, and z evaluated at u2 for the first curve section are

equal, respectively, to the values of x, y, and z evaluated at u1 of the next curve section.



• C1 continuity (First-order parametric continuity):

– Means that is the first parametric derivatives (tangent line) of the coordinate functions (above equations in slide 7) for two successive curve sections are equal at their joining point.

• C1 continuity (First-order parametric continuity):

– Means that is the first parametric derivatives (tangent line) of the coordinate functions (above equations in slide 7) for two successive curve sections are equal at their joining point.



• C2 continuity (Second-order parametric continuity):

– Means that both the first and second parametric derivatives of the two curve sections are the same at the intersection.

• Higher order parametric continuity conditions are defined similarly.

• C2 continuity (Second-order parametric continuity):

– Means that both the first and second parametric derivatives of the two curve sections are the same at the intersection.

• Higher order parametric continuity conditions are defined similarly.


Cubic Spline

• Cubic spline requires less calculations and storage space, and more stable than higher-order polynomials.

• Given a set of control points, cubic interpolation splines are obtained by fitting the input points with a piecewise cubic polynomial curve that passes through every control point.

• Cubic spline requires less calculations and storage space, and more stable than higher-order polynomials.

• Given a set of control points, cubic interpolation splines are obtained by fitting the input points with a piecewise cubic polynomial curve that passes through every control point.


Cubic Polynomials

• A parametric cubic polynomial that is to be fitted between each pair of control points with the following equations:

x(t) = axt3 + bxt2 + cxt + dx

y(t) = axt3 + bxt2 + cxt + dx

z(t) = axt3 + bxt2 + cxt + dx

0 <= t <= 1

• A parametric cubic polynomial that is to be fitted between each pair of control points with the following equations:

x(t) = axt3 + bxt2 + cxt + dx

y(t) = axt3 + bxt2 + cxt + dx

z(t) = axt3 + bxt2 + cxt + dx

0 <= t <= 1


Cubic Polynomials

Let T = [t3 t2 t 1]

Coefficient Matrix C

The the cuve Q(t) can be represented as:

Q(t) = T*C

Let T = [t3 t2 t 1]

Coefficient Matrix C

The the cuve Q(t) can be represented as:

Q(t) = T*C

z

z

y

y

x

x

zyx

zyx

d

c

d

c

d

c

bbb

aaa

ttt 123


Parametric CurvesHow do we find the tangent to a

curve?

• If f(x) = x2 -4

– tangent at (x=3) is

f’(x) = 2x - 4 = 2(3) - 4 = 2

How do we find the tangent to a curve?

• If f(x) = x2 -4

– tangent at (x=3) is

f’(x) = 2x - 4 = 2(3) - 4 = 2

Derivative of Q(t) is the tangent vector at t:Derivative of Q(t) is the tangent vector at t:

CttCTt

tQtQt

0123)(')( 2


Piecewise Curve Segments

One curve constructed by connecting many smaller segments end-to-end

• Must have rules for how the segments are joined

Continuity describes the joint

• Parametric continuity

• Geometric continuity

One curve constructed by connecting many smaller segments end-to-end

• Must have rules for how the segments are joined

Continuity describes the joint

• Parametric continuity

• Geometric continuity



• CC11 is tangent continuity (velocity) is tangent continuity (velocity)

• CC22 is 2 is 2ndnd derivative continuity (acceleration) derivative continuity (acceleration)

• Matching direction and magnitude of dMatching direction and magnitude of dnn / dt / dtnn

CCnn continous continous

• CC11 is tangent continuity (velocity) is tangent continuity (velocity)

• CC22 is 2 is 2ndnd derivative continuity (acceleration) derivative continuity (acceleration)

• Matching direction and magnitude of dMatching direction and magnitude of dnn / dt / dtnn

CCnn continous continous


Geometric Continuity

• Requires the parametric dervatives of the two sections are prportional to each other at their common boundary, instead of requiring equality.

• If positions match• G0 geometric continuity

• If direction (but not necessarily magnitude) of tangent matches

• G1 geometric continuity

• The tangent value at the end of one curve is proportional to the tangent value of the beginning of the next curve

• Requires the parametric dervatives of the two sections are prportional to each other at their common boundary, instead of requiring equality.

• If positions match• G0 geometric continuity

• If direction (but not necessarily magnitude) of tangent matches

• G1 geometric continuity

• The tangent value at the end of one curve is proportional to the tangent value of the beginning of the next curve


Parametric Cubic Curves

• In order to assure C2 continuity, curves must be of

at least degree 3

• Here is the parametric definition of a cubic (degree 3) spline in two dimensions

• How do we extend it to three dimensions?

• In order to assure C2 continuity, curves must be of

at least degree 3

• Here is the parametric definition of a cubic (degree 3) spline in two dimensions

• How do we extend it to three dimensions?


Parametric Cubic Splines

Can represent this as a matrix.Can represent this as a matrix.


Coefficients

So how do we select the coefficients?

• The coefficients [ax bx cx dx] and [ay by cy dy] must satisfy the

constraints defined by the continuity conditions

So how do we select the coefficients?

• The coefficients [ax bx cx dx] and [ay by cy dy] must satisfy the

constraints defined by the continuity conditions


Parametric Curves

• There are many splines developed, some of these are:

–Hermite

–Bezier

• There are many splines developed, some of these are:

–Hermite

–Bezier


Hermite Splines

• A Hermite spline is an interpolating cubic polynomial with a specified tangent at each control point.

• It can be adjusted locally because each curve section is only dependent on its endpoint constraints.

• A Hermite spline is an interpolating cubic polynomial with a specified tangent at each control point.

• It can be adjusted locally because each curve section is only dependent on its endpoint constraints.


Hermite Cubic Splines

• One cubic curve for each dimension.

• A curve constrained to x/y-plane has two curves.

• One cubic curve for each dimension.

• A curve constrained to x/y-plane has two curves.

d

c

b

a

ttt

dctbtattf x

1

)(

23

23

h

g

f

e

ttt

hgtftettf y

1

)(

23

23


Hermite Cubic Splines• A 2-D Hermite Cubic Spline is defined by eight parameters:

a, b, c, d, e, f, g, h

• How do we convert the intuitive endpoint constraints into these (relatively) unintuitive eight parameters?

• We know:• (x, y) position at t = 0, p(x, y) position at t = 0, p11

• (x, y) position at t = 1, p(x, y) position at t = 1, p22

• (x, y) derivative at t = 0, dp/dt(x, y) derivative at t = 0, dp/dt


• A 2-D Hermite Cubic Spline is defined by eight parameters: a, b, c, d, e, f, g, h

• How do we convert the intuitive endpoint constraints into these (relatively) unintuitive eight parameters?

• We know:• (x, y) position at t = 0, p(x, y) position at t = 0, p11

• (x, y) position at t = 1, p(x, y) position at t = 1, p22




Hermite Cubic Spline

We know:

• (x, y) position at t = 0, p1

We know:


xpdf

d

c

b

adcbaf

x

x

1

23

23

)0(

1000

000)0(

yphf

h

g

f

ehgfef

y

y

1

23

23

)0(

1000

000)0(



We know:


We know:


xpdcbaf

d

c

b

adcbaf

x

x

2

23

23

)1(

1111

111)1(

yphgfef

h

g

f

ehgfef

y

y

2

23

23

)1(

1111

111)1(


Hermite Cubic Splines

• So far we have four equations, but we have eight unknowns

• Use the derivatives

• So far we have four equations, but we have eight unknowns

• Use the derivatives

d

c

b

a

tttf

cbtattf

dctbtattf

x

x

x

0123)(

23)(

)(

2

2

23

h

g

f

e

tttf

gftettf

hgtftettf

y

y

y

0123)(

23)(

)(

2

2

23



We know:

• (x, y) derivative at t = 0, dp/dt

We know:


dtdp

cf

d

c

b

acbaf

xx

x

1

2

2

)0(

010203

0203)0(

dtdp

gf

h

g

f

egfef

y

y

y

1

2

2

)0(

010203

0203)0(



We know:


We know:


dtdp

cbaf

d

c

b

acbaf

xx

x

2

2

2

23)1(

011213

1213)1(

dtdp

gfef

h

g

f

egfef

y

y

y

2

2

2

23)1(

011213

1213)1(


Hermite Specification

• Matrix equation for Hermite Curve• Matrix equation for Hermite Curve

t = 0

t = 1

t = 0

t = 1

t3 t2 t1 t0

p1

p2

r p1

r p2

dtdp

dtdp

p

p

dtdp

dtdpp

p

h

g

f

e

d

c

b

a

y

y

y

y

x

x

x

x

2

1

2

1

2

1

2

1

0123

0100

1111

1000


Solve Hermite Matrix

h

g

f

e

d

c

b

a

dtdp

dtdp

p

p

dtdp

dtdpp

p

y

y

y

y

x

x

x

x

2

1

2

1

2

1

2

11

0123

0100

1111

1000


Spline and Geometry Matrices

h

g

f

e

d

c

b

a

dtdp

dtdp

p

p

dtdp

dtdpp

p

y

y

y

y

x

x

x

x

2

1

2

1

2

1

2

1

0001

0100

1233

1122

MHermite GHermite


Resulting Hermite Spline Equation


Sample Hermite Curves


Hermite Curves

• If P(u) represents a parametric cubic point functionfor the cuve section between control points Pk and Pk+1 as shown bellow:

• If P(u) represents a parametric cubic point functionfor the cuve section between control points Pk and Pk+1 as shown bellow:

Pk

Pk+1

P(u) = (x(u), y(u), z(u)), where

• x(u)=axu3 + bxu2 + cxu + dx

• y(u)=ayu3 + byu2 + cyu + dy

• z(u)=azu3 + bzu2 + czu + dz


Hermite Curves

• The boundary conditions that define this Hermite curve section are:

P(0) = Pk

P(1) = Pk+1

P’(0) = DPk

P’(1) = DPk+1

• With DPk and DPk+1 specifying the values for the parametric derivatives at control points Pk and Pk+1

• The boundary conditions that define this Hermite curve section are:

P(0) = Pk

P(1) = Pk+1

P’(0) = DPk

P’(1) = DPk+1

• With DPk and DPk+1 specifying the values for the parametric derivatives at control points Pk and Pk+1


Hermite Curves

• We can write the vector equivalent for the Hermit curve section as:

– P(u) = a u3 + b u2 + cu + d 0<= u <= 1

• Where the x, y and z components of P(u) is

x(u)=axu3 + bxu2 + cxu + dx

y(u)=ayu3 + byu2 + cyu + dy

z(u)=azu3 + bzu2 + czu + dz

• We can write the vector equivalent for the Hermit curve section as:

– P(u) = a u3 + b u2 + cu + d 0<= u <= 1

• Where the x, y and z components of P(u) is

x(u)=axu3 + bxu2 + cxu + dx

y(u)=ayu3 + byu2 + cyu + dy

z(u)=azu3 + bzu2 + czu + dz


Hermite Curves

• The matrix equivalent is

• And the derivative of the point function is

• The matrix equivalent is

• And the derivative of the point function is

d

c

b

a

uuuuP 1)( 23

d

c

b

a

uuuP 0123)(' 2


Hermite Curves

• Substituting endpoint values 0 and 1 for parameter u into the above two equations, we can express the Hermite boundary conditions in the matrix form

• Substituting endpoint values 0 and 1 for parameter u into the above two equations, we can express the Hermite boundary conditions in the matrix form

d

c

b

a

D

D

P

P

PK

PK

k

k

0123

0100

1111

1000

1

1


Hermite Curves

• Solving this equation for the polynomial coefficients, we have

• MH, the Hermite matrix, is the inverse of the boundary condition matrix.

• Solving this equation for the polynomial coefficients, we have

• MH, the Hermite matrix, is the inverse of the boundary condition matrix.

1

1

1

0123

0100

1111

1000

PK

PK

k

k

D

D

P

P

d

c

b

a

1

1

1

0001

0100

1233

1122

PK

PK

k

k

D

D

P

P

1

1

PK

PK

k

k

H

D

D

P

P

M


Hermite Curves

• The above matrix can be written in terms of the boundary conditions as

• This can be used to determine the expressions for the polynomial Herimte functions.

• The above matrix can be written in terms of the boundary conditions as

• This can be used to determine the expressions for the polynomial Herimte functions.

1

123 1)(

PK

PK

k

k

H

D

D

P

P

MuuuuP


Hermite Blending Functions

• The above matrix can be used to determine the expressions for the polynomial Herimte functions,

– Hk(u) for k=0,1,2,3

• This can be done by carrying out the matrix multiplications and collecting coefficient for the boundary conditions to obtain the polynomial form:

• P(u) = Pk(2u3 - 3u2 + 1) + Pk+1(-2u3 + 3u2) + DPk(u3 -2u2 + u) + DPk+1(u3 u2)

• = Pk H0(u) + Pk+1 H1(u) + DPk H2(u) + DPk+1 H3(u)

• The above matrix can be used to determine the expressions for the polynomial Herimte functions,

– Hk(u) for k=0,1,2,3

• This can be done by carrying out the matrix multiplications and collecting coefficient for the boundary conditions to obtain the polynomial form:

• P(u) = Pk(2u3 - 3u2 + 1) + Pk+1(-2u3 + 3u2) + DPk(u3 -2u2 + u) + DPk+1(u3 u2)

• = Pk H0(u) + Pk+1 H1(u) + DPk H2(u) + DPk+1 H3(u)


Blending Functions