CENG 477 Introduction to Computer Graphicssaksagan.ceng.metu.edu.tr/courses/ceng477/files/pdf/week_12.pdfBezier Curves • Bezier curves are related to the Hermite curves as: • The

CENG 477Introduction to Computer Graphics

Representing Curves and Surfaces

Introduction• There are no straight lines or flat faces in nature!• Therefore, representing and generating smooth shapes is a

requirement in many CG applications• Rendering will still use lines and triangles but their vertices

will be sampled from a curve or surface

CENG 477 – Computer Graphics 2

Curves• There are many ways to represent curves:

– must be practical (easy to manage and render)– must be flexible (general enough to be used in various modeling tasks)

• A good compromise is cubic polynomials• Each x, y, z coordinate expressed a cubic polynomial with

potentially different coefficients


𝑥 𝑡 = 𝑎&𝑡' + 𝑏&𝑡* + 𝑐&𝑡 + 𝑑&

𝑦 𝑡 = 𝑎.𝑡' + 𝑏.𝑡* + 𝑐.𝑡 + 𝑑.

𝑧 𝑡 = 𝑎0𝑡' + 𝑏0𝑡* + 𝑐0𝑡 + 𝑑0

Curves• For a given t value, 𝑞 𝑡 = [𝑥 𝑡 𝑦 𝑡 𝑧 𝑡 ]represents the 3D

position along the curve• Almost like ray tracing except that the ray follows a curvy path

instead of a straight one!• The 𝑡 parameter is taken to be in range [0, 1]


𝑞(0)𝑞(1)

𝑞(0.5)

Cubic Polynomials• As cubic polynomials have 4 unknowns (per component), we

need 4 constraints to find them• Different curves are distinguished by different constraints

– Hermite curves: 2 end points + 2 tangent vectors– Bezier curves: 2 end points + 2 control points– Splines: 4 control points (for each piece of the curve)


Matrix Form• Cubic polynomials are conveniently expressed in matrix form:


𝑥 𝑡 = 𝑎&𝑡' + 𝑏&𝑡* + 𝑐&𝑡 + 𝑑&

𝑦 𝑡 = 𝑎.𝑡' + 𝑏.𝑡* + 𝑐.𝑡 + 𝑑.

𝑧 𝑡 = 𝑎0𝑡' + 𝑏0𝑡* + 𝑐0𝑡 + 𝑑0

𝑄 𝑡 = 𝑥 𝑡 𝑦 𝑡 𝑧 𝑡 = [𝑡'𝑡*𝑡1]

𝑎& 𝑎. 𝑎0𝑏& 𝑏. 𝑏0𝑐& 𝑐. 𝑐0𝑑& 𝑑. 𝑑0

𝑄 𝑡 = 𝑇𝐶

Matrix Form• We will also need the derivative of this curve to specify

tangent vectors


𝑑𝑄 𝑡𝑑𝑡 = 𝑄> 𝑡 =

𝑑𝑇𝑑𝑡 𝐶 = 3𝑡*2𝑡10 𝐶

Constraints• Imagine that we want to specify certain geometrical

constraints such as:– Start point– End point– Start direction (i.e. tangent vector at start point)– End direction (i.e. tangent vector at end point)

• We need to split the matrix C into two to allow embedding these constraints


Constraints• Rewrite 𝐶 = 𝑀𝐺, where 𝐺 represents the geometry constraints

• Here, 𝑀 is called the basis matrix• 𝐺 is called the geometry or the constraints matrix• Different types of curves differ in 𝑀 and 𝐺


C =

𝑎& 𝑎. 𝑎0𝑏& 𝑏. 𝑏0𝑐& 𝑐. 𝑐0𝑑& 𝑑. 𝑑0

=

𝑚DD 𝑚D* 𝑚D' 𝑚DE𝑚*D 𝑚** 𝑚*' 𝑚*E𝑚'D 𝑚'* 𝑚'' 𝑚'E𝑚ED 𝑚E* 𝑚E' 𝑚EE

𝑔D& 𝑔D. 𝑔D0𝑔*& 𝑔*. 𝑔*0𝑔'& 𝑔'. 𝑔'0𝑔E& 𝑔E. 𝑔E0

𝑀 𝐺

Constraints• Note that 𝑄 𝑡 = 𝑇𝑀𝐺• For instance:


𝑥 𝑡 = [𝑡'𝑡*𝑡1] 𝑔D&

𝑚DD𝑚*D𝑚'D𝑚ED

+ 𝑔*&

𝑚D*𝑚**𝑚'*𝑚E*

+ 𝑔'&

𝑚D'𝑚*'𝑚''𝑚E'

+ 𝑔E&

𝑚DE𝑚*E𝑚'E𝑚EE

Blending Functions• Rewriting this gives us:

• That is, the curve is a weighted sum of the elements of the geometry matrix


𝑥 𝑡 = [𝑡'𝑡*𝑡1] 𝑔D&

𝑚DD𝑚*D𝑚'D𝑚ED

+ 𝑔*&

𝑚D*𝑚**𝑚'*𝑚E*

+ 𝑔'&

𝑚D'𝑚*'𝑚''𝑚E'

+ 𝑔E&

𝑚DE𝑚*E𝑚'E𝑚EE

𝑥 𝑡 = 𝑡'𝑚DD + 𝑡*𝑚*D + 𝑡𝑚'D +𝑚ED 𝑔D& +𝑡'𝑚D* + 𝑡*𝑚** + 𝑡𝑚'* +𝑚E* 𝑔*& +𝑡'𝑚D' + 𝑡*𝑚*' + 𝑡𝑚'' +𝑚E' 𝑔'& +𝑡'𝑚DE + 𝑡*𝑚*E + 𝑡𝑚'E +𝑚EE 𝑔E&

Blending Functions• The weights are each cubic polynomials of 𝑡• These polynomials are called blending functions


𝑥 𝑡 = 𝑡'𝑚DD + 𝑡*𝑚*D + 𝑡𝑚'D +𝑚ED 𝑔D& +𝑡'𝑚D* + 𝑡*𝑚** + 𝑡𝑚'* +𝑚E* 𝑔*& +𝑡'𝑚D' + 𝑡*𝑚*' + 𝑡𝑚'' +𝑚E' 𝑔'& +𝑡'𝑚DE + 𝑡*𝑚*E + 𝑡𝑚'E +𝑚EE 𝑔E&

𝐵D&E = 𝑇D&E𝑀E&E

Vector of blending functions

Hermite Curves• The constraints of Hermite curves are:

– Two end points: 𝑃D and 𝑃E– Two tangent vectors: 𝑅D and 𝑅E


𝑃D

𝑃E𝑅D

𝑅E

Hermite Curves• The geometry matrix then becomes:

• To find 𝑀 remember that the curve in matrix form we have:

• And for derivative:


𝐺 =

𝑃D𝑃E𝑅D𝑅E

=

𝑃D& 𝑃D. 𝑃D0𝑃E& 𝑃E. 𝑃E0𝑅D& 𝑅D. 𝑅D0𝑅E& 𝑅E. 𝑅E0

𝑄 𝑡 = 𝑇𝑀𝐺

𝑄′ 𝑡 = 𝑇′𝑀𝐺

Hermite Curves• We can now plug in values for the 𝑡 parameter

– Compute 𝑄 0 ,𝑄 1 ,𝑄> 0 , 𝑄′(1)

• This is the same as:


𝑄 0 = [0001]𝑀𝐺𝑄 1 = [1111]𝑀𝐺𝑄′ 0 = [0010]𝑀𝐺𝑄′ 1 = [3210]𝑀𝐺

𝑄(0)𝑄(1)𝑄′(0)𝑄′(1)

=

0 0 0 11 1 1 10 0 1 03 2 1 0

𝑀𝐺

Hermite Curves• Remember that 𝐺 was equal to:

• So we have:


𝐺 =


=

𝑄(0)𝑄(1)𝑄′(0)𝑄′(1)

𝑄(0)𝑄(1)𝑄′(0)𝑄′(1)

=

0 0 0 11 1 1 10 0 1 03 2 1 0

𝑀

𝑄(0)𝑄(1)𝑄′(0)𝑄′(1)

Same

Only possibleif 𝑀 is the inverseof the matrix

Hermite Curves• Therefore, Hermite curves have the following basis matrix:

• This yields the following blending functions:


𝑀 =

0 0 0 11 1 1 10 0 1 03 2 1 0

KD

=

2 −2 1 1−3 3 −2 −10 0 1 01 0 0 0

𝐵 = 2𝑡' − 3𝑡* + 1−2𝑡' + 3𝑡*𝑡' − 2𝑡* + 𝑡𝑡' − 𝑡*

M Let’s plot thesewith gnuplot anddo some experimentationwith Matlab!

Hermite Blending Functions


Bezier Curves• Bezier curves can draw the same curves as Hermite curves• They are defined using control points instead of derivatives• Two control points are interpolated and two control points are

approximated


Bezier Curves• The curve lies entirely within the convex-hull of these four

points• This is useful for clipping, culling, and intersection tests as the

convex-hull can be tested first instead of each line segment


Bezier Curves• Bezier curves are related to the Hermite curves as:

• The factor 3 ensures that 𝑃* has the highest weight at 𝑡 = 1/3and 𝑃' has the highest weight at 𝑡 = 2/3, logically dividing the curve into 3 pieces


𝑅D = 3(𝑃* − 𝑃D)

𝑅E = 3(𝑃E − 𝑃')

Bezier Curves• In matrix form, this relationship can be express as:


𝐺O =


=

1 0 0 00 0 0 1−3 3 0 00 0 −3 3

𝑃D𝑃*𝑃'𝑃E

𝐺P

This matrix translates Bezier geometrymatrix to the Hermite geometry matrix(let’s call this matrix as 𝑀PO)

Bezier Curves• Then Bezier curves can be defined as:


𝑄 𝑡 = 𝑇𝑀O𝐺O = 𝑇𝑀O𝑀PO𝑀P

𝑄 𝑡 = 𝑇𝑀P𝐺P

𝑀P = 𝑀O𝑀POwhere

𝑄 𝑡 = 𝑡' 𝑡* 𝑡 1

−1 3 −3 13 −6 3 0−3 3 0 01 0 0 0


Bezier Curves• This is equivalent to:

• These are called Bernstein polynomials– Their sum is always 1– They are always non-negative when 𝑡 ∈ [0,1]– That’s why the resulting curve is in the convex-hull of 𝑃D, 𝑃*, 𝑃', 𝑃E– Bernstein polynomial of degree 𝑛 is 𝐵T,U 𝑡 = 𝑛𝑖 𝑡T 1 − 𝑡 UKT


𝑄 𝑡 = −𝑡' + 3𝑡* − 3𝑡 + 1 𝑃D +

3𝑡' − 6𝑡* + 3𝑡 𝑃* +−3𝑡' + 3𝑡* 𝑃' +𝑡' 𝑃E

=(1 − 𝑡)'𝑃D +3𝑡(1 − 𝑡)*𝑃* +3𝑡*(1 − 𝑡)𝑃' +𝑡'𝑃E +

Bezier Curves• Bezier curve is the sum of Bernstein polynomials


𝑄 𝑡 = (1 − 𝑡)'𝑃D + 3𝑡(1 − 𝑡)*𝑃* + 3𝑡*(1 − 𝑡)𝑃' + 𝑡'𝑃E

𝑄 𝑡 = W𝐵T,U 𝑡 𝑃TXD

U

TYZ

𝐵T,U 𝑡 = 𝑛𝑖 𝑡T 1 − 𝑡 UKT

Bernstein Polynomials


Continuity• Until now, we learned to draw a single curve segment• If we want to combine multiple curve segments, we must

ensure maintaining continuity


Curve 1 Curve 2

Types of Continuity• No continuity:

– The curves do not meet

• C0 continuity:– The end points meet, also know as positional continuity


Types of Continuity• C1 continuity:

– The curves meet and have identical tangent vectors at the connection

• C2 continuity:– The curves meet and have identical curvature at the connection– The curvature is defined as the rate of change of tangents


Types of Continuity• Imagine a camera moving along a curve with multiple segments

– No continuity: camera will make jumps between segments– C0 continuity: camera velocity may suddenly change– C1 continuity: camera acceleration may suddenly change– C2 continuity: camera motion will appear smooth

• In general, maintaining C2 continuity is desired


Maintaining Continuity• Imagine having two Hermite curves:

• C1 continuity can be maintained if 𝑃E = 𝑄D and 𝑅E = 𝑇D


𝐺[ =


𝐺\ =

𝑄D𝑄E𝑇D𝑇E

Maintaining Continuity• Similarly for two Bezier curves:

• C1 is ensured if 𝑃E = 𝑄D and 𝑃E − 𝑃' = 𝑄* − 𝑄D


𝐺[ =


𝐺\ =

𝑄D𝑄*𝑄'𝑄E

Maintaining Continuity• What if we want to maintain C2 continuity?• Unfortunately, neither Hermite nor Bezier curves can

guarantee C2 continuity• For this we have a new type of curve called splines


Splines• The term spline was used to refer to flexible metal strips used

by draftspersons to design the surfaces of airplanes, cars, and ships, etc.


core77.com

Splines• The splines, due to physical properties of the metal strips, had

second order (C2) continuity• Its mathematical equivalent is natural cubic splines• Splines have one more degree of continuity than that is

afforded by Hermite and Bezier curves• There are other types of splines:

– B-Splines– Uniform Nonrational B-Splines– Nonuniform Nonrational B-Splines– Nonuniform Rational B-Splines– Beta-Splines– V-Splines


Natural Cubic Splines• Defined by n+1 control points• The spline, consisting of n curves, interpolates all of these

points


𝑃Z

𝑃D

𝑃*

𝑃'𝑃E

𝑃UKD

𝑃U

𝑆Z(𝑡)

𝑆D(𝑡)

𝑆*(𝑡)

𝑆'(𝑡)

𝑆UKD(𝑡)

Natural Cubic Splines• The spline is defined as:

• Each curve is a cubic polynomial:


𝑆 𝑡 =

𝑆Z 𝑡 , 𝑡Z ≤ 𝑡 ≤ 𝑡D𝑆D 𝑡 , 𝑡D ≤ 𝑡 ≤ 𝑡*⋮

𝑆UKD 𝑡 , 𝑡UKD ≤ 𝑡 ≤ 𝑡U

𝑆Z 𝑡 = 𝑎Z𝑡' + 𝑏Z𝑡* + 𝑐Z𝑡 + 𝑑Z⋮

𝑆UKD 𝑡 = 𝑎UKD𝑡' + 𝑏UKD𝑡* + 𝑐UKD𝑡 + 𝑑UKD

There are a totalof 4n unknowns!

Natural Cubic Splines• The end points of the curves must meet (C0 cont.) and their

first two derivatives must be equal (C1 and C2):

• This gives us 3n – 3 equations


𝑆TKD 𝑡T = 𝑆T(𝑡T)𝑆TKD> 𝑡T = 𝑆T>(𝑡T)

𝑆TKD>> 𝑡T = 𝑆T>>(𝑡T)𝑖 = 1…𝑛 − 1

Natural Cubic Splines• We also know the values of the spline at the control points:

• This gives us another n+1 equations• We still need two more …• 𝑆>′ 𝑡Z = 𝑆>′ 𝑡U = 0 gives us natural cubic splines


𝑆 𝑡T = 𝑃T, 𝑖 = 0… 𝑛

Computing Polynomials• Let’s call the second derivatives at control points as:

• For natural cubic spline we have 𝑍Z = 𝑍U = [000]M• How does 𝑆′′(𝑡) look like?


𝑍T = 𝑆>>(𝑡T)

𝑆′(𝑡) 𝑆′′(𝑡)𝑆(𝑡)

Computing Polynomials• 𝑆′′(𝑡) will be piecewise-linear


𝑍Z = 𝟎M

𝑍*

𝑍'

𝑍E

𝑍UKD

𝑍U = 𝟎M

𝑡Z 𝑡* 𝑡' 𝑡E 𝑡UKD 𝑡U

𝑆T>> 𝑡 = 𝑍T𝑡 − 𝑡TXD𝑡T − 𝑡TXD

+𝑍TXD𝑡 − 𝑡T𝑡TXD − 𝑡T𝑍D

𝑡D

This derivation is largely inspired from Arne Morten Kvarving’s slides on cubic splines

Computing Polynomials• 𝑆′′(𝑡) will be piecewise-linear


𝑍Z = 𝟎M

𝑍*

𝑍'

𝑍E

𝑍UKD

𝑍U = 𝟎M

𝑡Z 𝑡* 𝑡' 𝑡E 𝑡UKD 𝑡U

𝑆T>> 𝑡 = 𝑍TXD𝑡 − 𝑡TℎT

+ 𝑍T𝑡TXD − 𝑡ℎT𝑍D

𝑡D

where ℎT = 𝑡TXD − 𝑡T

Computing Polynomials• At this point, we need to integrate twice to obtain 𝑆(𝑡)


𝑆T>> 𝑡 = 𝑍TXD𝑡 − 𝑡TℎT

+ 𝑍T𝑡TXD − 𝑡ℎT

𝑆T> 𝑡 = 𝑍TXD(𝑡 − 𝑡T)*

2ℎT+ 𝑍T

(𝑡TXD − 𝑡)*

−2ℎT+ 𝐶T

𝑆T 𝑡 = 𝑍TXD(𝑡 − 𝑡T)'

6ℎT+ 𝑍T

(𝑡TXD − 𝑡)'

6ℎT+ 𝐶T𝑡 + 𝐷T


6ℎT+ 𝑍T

(𝑡TXD − 𝑡)'

6ℎT+ 𝐸T(𝑡 − 𝑡T) + 𝐹T(𝑡TXD − 𝑡)

where𝐶T = 𝐸T − 𝐹T and 𝐷T = 𝐹T𝑡TXD − 𝐸T𝑡T

Computing Polynomials• At this point, the only unknowns are 𝑍T, 𝑍TXD, 𝐸T and 𝐹T

• Plug-in the values at the control points:

• From here, we can determine 𝐸T and 𝐹T



6ℎT+ 𝑍T

(𝑡TXD − 𝑡)'

6ℎT+ 𝐸T(𝑡 − 𝑡T) + 𝐹T(𝑡TXD − 𝑡)

𝑆T 𝑡T = 𝑃T = 𝑍TℎT*

6 + 𝐹TℎT

𝑆T 𝑡TXD = 𝑃TXD = 𝑍TXDℎT*

6 + 𝐸TℎT

Computing Polynomials• This gives us:

• Finally, we need to compute the 𝒁𝒊 terms• We know that 𝒁𝟎 = 𝒁𝒏 = [000]M


𝑆T 𝑡 = 𝒁𝒊X𝟏(𝑡 − 𝑡T)'

6ℎT+ 𝒁𝒊

(𝑡TXD − 𝑡)'

6ℎT+

𝑃TXDℎT

−𝑍TXDℎT6 (𝑡 − 𝑡T) +

𝑃TℎT−𝑍TℎT6 (𝑡TXD − 𝑡)

Computing Polynomials• We did not use the constraint that the first derivatives at the

control points are equal; so take the derivative


𝑺𝒊> 𝒕 = 𝒁𝒊X𝟏(𝑡 − 𝑡T)*

2ℎT− 𝒁𝒊

𝑡TXD − 𝑡 *

2ℎT+1ℎT

𝑷𝒊X𝟏 − 𝑷𝒊 −ℎT6 𝒁𝒊X𝟏 − 𝒁𝒊

𝑩𝒊

𝑺𝒊> 𝒕𝒊 = −𝒁𝒊ℎT2 +𝑩𝒊 −

ℎT6 𝒁𝒊X𝟏 +

ℎT6 𝒁𝒊

Computing Polynomials• Repeat this for the previous (or the next) segment:


𝑺𝒊K𝟏> 𝒕 = 𝒁𝒊(𝑡 − 𝑡TKD)*

2ℎTKD− 𝒁𝒊K𝟏

𝑡T − 𝑡 *

2ℎTKD+

1ℎTKD

𝑷𝒊 − 𝑷𝒊K𝟏 −ℎTKD6 𝒁𝒊 − 𝒁𝒊K𝟏

𝑩𝒊K𝟏

𝑺𝒊K𝟏> 𝒕𝒊 = 𝒁𝒊ℎTKD2 +𝑩𝒊K𝟏 −

ℎTKD6 𝒁𝒊 +

ℎTKD6 𝒁𝒊K𝟏

Computing Polynomials• Now equate the segments at the control points:


−𝒁𝒊ℎT2 +𝑩𝒊 −

ℎT6 𝒁𝒊X𝟏 +

ℎT6 𝒁𝒊 = 𝒁𝒊

ℎTKD2 +𝑩𝒊K𝟏 −

ℎTKD6 𝒁𝒊 +

ℎTKD6 𝒁𝒊K𝟏

𝑺𝒊> 𝒕𝒊 = 𝑺𝒊K𝟏> 𝒕𝒊

−3𝒁𝒊ℎT + 6𝑩𝒊 − ℎT𝒁𝒊X𝟏 + ℎT𝒁𝒊 = 3𝒁𝒊ℎTKD + 6𝑩𝒊K𝟏 − ℎTKD𝒁𝒊 + ℎTKD𝒁𝒊K𝟏

6(𝑩𝒊 − 𝑩𝒊K𝟏) = ℎTKD𝒁𝒊K𝟏 +2(ℎTKD + ℎT)𝒁𝒊+ℎT𝒁𝒊X𝟏

Computing Polynomials• We can set up n-1 equations in this form and we also have n-1

unknowns (from 𝑍D to 𝑍UKD)

• In the above equation, plug 𝑖 = 1…𝑛 − 1, and solve the resulting system


6(𝑩𝒊 − 𝑩𝒊K𝟏) = ℎTKD𝒁𝒊K𝟏 +2(ℎTKD + ℎT)𝒁𝒊+ℎT𝒁𝒊X𝟏

Computing Polynomials• Setup the system such that we have the 𝐴𝑥 = 𝑏 form


𝑣D ℎD 0 …ℎD 𝑣* ℎ*0 ℎ* 𝑣' ℎ'⋮ ⋱

𝑣UK* ℎUK*ℎUK* 𝑣UKD

𝑍D,&𝑍*,&𝑍',&⋮

𝑍UKD,&

=

𝑏D𝑏*𝑏'⋮

𝑏UKD

𝑣T = 2(ℎTKD + ℎT)

𝑏T = 6(𝐵T,& − 𝐵TKD,&)

• Note that we are solving for the x-components• We need to solve for y- and z-components if

our curve is 3 dimensional

ℎT = 𝑡TXD − 𝑡T

𝐵T,& =1ℎT

𝑃TXD,& − 𝑃T,&

Sample Output


P = {(1, 1), (2, 3), (3, 2), (4, 5), (5, 1), (6, 3), (7, 2)}T = {0, 1, 2, 3, 4, 5, 6}

Parametric Bicubic Surfaces• Generalization of parametric cubic curves• Recall 𝑄 𝑡 = 𝑇𝑀𝐺• First replace 𝑡 by 𝑠 such that 𝑄 𝑠 = 𝑆𝑀𝐺• Now allow the points in 𝐺 to vary along a curve parametrized

by 𝑡


𝑄 𝑠, 𝑡 = 𝑆𝑀𝐺(𝑡)

𝑄 𝑠, 𝑡 = 𝑆𝑀

𝐺D(𝑡)𝐺*(𝑡)𝐺'(𝑡)𝐺E(𝑡)

𝐺D&(𝑡) 𝐺D.(𝑡) 𝐺D0(𝑡)

Parametric Bicubic Surfaces• We can setup separate equations for x, y, and z:


𝑄& 𝑠, 𝑡 = 𝑆𝑀

𝐺D&(𝑡)𝐺*&(𝑡)𝐺'&(𝑡)𝐺E&(𝑡)

𝑄. 𝑠, 𝑡 = 𝑆𝑀

𝐺D.(𝑡)𝐺*.(𝑡)𝐺'.(𝑡)𝐺E.(𝑡)

𝑄0 𝑠, 𝑡 = 𝑆𝑀

𝐺D0(𝑡)𝐺*0(𝑡)𝐺'0(𝑡)𝐺E0(𝑡)

Parametric Bicubic Surfaces• Now for a fixed 𝑡 = 𝑡D, 𝑄(𝑠, 𝑡D) is a curve because 𝐺(𝑡D) is

constant• Allowing 𝑡 to take on a different value, 𝑡*, where 𝑡* − 𝑡D is

very small, 𝑄(𝑠, 𝑡*) is a slightly different curve• Repeating this arbitrarily many times gives you a large set of

curves, which is our surface


…

Called bicubic if 𝐺T (𝑡) are themselves cubic

Derivation• Assume that 𝐺T(𝑡) themselves are defined by:


𝐺T 𝑡 = 𝑇𝑀𝑍T

𝐺T 𝑡 = 𝑇𝑀

𝑍TD𝑍T*𝑍T'𝑍TE

𝑍TD& 𝑍TD. 𝑍TD0

𝐺T& 𝑡 = 𝑇𝑀

𝑍TD&𝑍T*&𝑍T'&𝑍TE&

𝐺T. 𝑡 = 𝑇𝑀

𝑍TD.𝑍T*.𝑍T'.𝑍TE.

𝐺T0 𝑡 = 𝑇𝑀

𝑍TD0𝑍T*0𝑍T'0𝑍TE0

Derivation• Remember that we have:

• To combine them into a single equation, take the transpose of 𝐺T& 𝑡 , which is equal to itself due to its being a scalar:






𝐺D& 𝑡 M = 𝐺D& 𝑡 = 𝑍DD& 𝑍DD& 𝑍D'& 𝑍DE& 𝑀M𝑇M

𝐺*& 𝑡 M = 𝐺*& 𝑡 = 𝑍*D& 𝑍*D& 𝑍*'& 𝑍*E& 𝑀M𝑇M

𝐺'& 𝑡 M = 𝐺'& 𝑡 = 𝑍'D& 𝑍'D& 𝑍''& 𝑍'E& 𝑀M𝑇M

𝐺E& 𝑡 M = 𝐺E& 𝑡 = 𝑍ED& 𝑍ED& 𝑍E'& 𝑍EE& 𝑀M𝑇M

Derivation• Remember that we have:

• To combine them into a single equation, take the transpose of 𝐺T& 𝑡 , which is equal to itself due to its being a scalar:






𝐺D& 𝑡𝐺*& 𝑡𝐺'& 𝑡𝐺E& 𝑡

=

𝑍DD& 𝑍DD& 𝑍D'& 𝑍DE&𝑍*D& 𝑍*D& 𝑍*'& 𝑍*E&𝑍'D& 𝑍'D& 𝑍''& 𝑍'E&𝑍ED& 𝑍ED& 𝑍E'& 𝑍EE&

𝑀M𝑇M

Derivation• This gives us:

• Similarly for y and z:


𝑄& 𝑠, 𝑡 = 𝑆

𝑚DD 𝑚D* 𝑚D' 𝑚DE𝑚*D 𝑚** 𝑚*' 𝑚*E𝑚'D 𝑚'* 𝑚'' 𝑚'E𝑚ED 𝑚E* 𝑚E' 𝑚EE

𝑍DD& 𝑍DD& 𝑍D'& 𝑍DE&𝑍*D& 𝑍*D& 𝑍*'& 𝑍*E&𝑍'D& 𝑍'D& 𝑍''& 𝑍'E&𝑍ED& 𝑍ED& 𝑍E'& 𝑍EE&

𝑀M𝑇M

𝑀 𝐺&

𝑄. 𝑠, 𝑡 = 𝑆𝑀𝐺.𝑀M𝑇M

𝑄0 𝑠, 𝑡 = 𝑆𝑀𝐺0𝑀M𝑇M

Hermite Surfaces• For Hermite surfaces, 𝑀 is the Hermite basis matrix• The elements of the geometry matrix (𝐺&, 𝐺., 𝐺0) store how

each component changes with respect to t:


…𝑃DD = 𝑄(0, 0)

𝑃DE = 𝑄(0, 1)

𝑅DD = 𝑑𝑄(0,0)/𝑑𝑠

𝑅DE = 𝑑𝑄(0,1)/𝑑𝑠

𝑃ED = 𝑄(1, 0)

𝑃EE = 𝑄(1, 1)

𝑅ED = 𝑑𝑄(1, 0)/𝑑𝑠

𝑅EE = 𝑑𝑄(1, 1)/𝑑𝑠

Hermite Surfaces• How the starting point (𝑃D) changes with respect to t:


𝐺& =𝑄&(0, 0) 𝑄&(0, 1)

𝑑𝑄&(0, 0)𝑑𝑡

𝑑𝑄&(0, 1)𝑑𝑡… … … …

… … … …… … … …

Hermite Surfaces• How the end point (𝑃E) changes with respect to t:


𝐺& =

… … … …

𝑄&(1, 0) 𝑄&(1, 1)𝑑𝑄&(1, 0)

𝑑𝑡𝑑𝑄&(1,1)

𝑑𝑡… … … …… … … …

Hermite Surfaces• How the starting tangent vector (𝑅D), defined with respect to 𝑠,

changes with respect to t:


𝐺& =

… … … …… … … …

𝑑𝑄&(0, 0)𝑑𝑠

𝑑𝑄&(0, 1)𝑑𝑠

𝑑*𝑄&(0, 0)𝑑𝑠𝑑𝑡

𝑑*𝑄&(0, 1)𝑑𝑠𝑑𝑡… … … …

Hermite Surfaces• How the ending tangent vector (𝑅E), defined with respect to 𝑠,

changes with respect to t:


𝐺& =

… … … …… … … …… … … …

𝑑𝑄&(1, 0)𝑑𝑠

𝑑𝑄&(1, 1)𝑑𝑠

𝑑*𝑄&(1,0)𝑑𝑠𝑑𝑡


Hermite Surfaces• So the entire geometry matrix looks like:


𝐺& =

𝑄&(0,0) 𝑄&(0,1)𝑑𝑄&(0, 0)

𝑑𝑡𝑑𝑄&(0, 1)

𝑑𝑡

𝑄&(1,0) 𝑄&(1,1)𝑑𝑄&(1, 0)

𝑑𝑡𝑑𝑄&(1, 1)

𝑑𝑡𝑑𝑄&(0, 0)

𝑑𝑠𝑑𝑄&(0, 1)

𝑑𝑠𝑑*𝑄&(0,0)𝑑𝑠𝑑𝑡


𝑑𝑄&(1, 0)𝑑𝑠

𝑑𝑄&(1, 1)𝑑𝑠

𝑑*𝑄&(1,1)𝑑𝑠𝑑𝑡


Hermite Surfaces


𝐺& =

0 5 5 50 5 5 50 0 0 00 0 0 0

𝐺. =

0 0 0 05 5 0 05 5 0 05 5 0 0

𝐺0 =

0 0 5 50 0 5 50 0 0 00 0 0 0

Bezier Surfaces• Remember that Bezier curves was defined using Bernstein

polynomials:

• Their extension to surfaces is straightforward:


𝑄 𝑡 = W𝐵T,U 𝑡 𝑃TXD

U

TYZ𝐵T,U 𝑡 = 𝑛𝑖 𝑡T 1 − 𝑡 UKT

𝑄 𝑠, 𝑡 =WW𝐵T,U 𝑠 𝐵T,s 𝑡 𝑃TXD

s

tYZ

U

TYZ

Bezier Surfaces


• Similar to a Bezier curve, a Bezier surface interpolates the end points and approximates the interior control points

Bezier Surfaces• Complex models can be created using Bezier surfaces• In such models, the entire surface is composed of multiple

Bezier surfaces, known as patches


Gumbo Model

Bezier Surfaces• Such patches allows tessellating a surface at the desired level

of detail depending on viewing distance or other parameters• OpenGL tessellation shaders provide hardware support for this


Documents

CENG 477 Introduction to Computer Graphicssaksagan.ceng.metu.edu.tr/courses/ceng477/files/pdf/week_12.pdfBezier Curves • Bezier curves are related to the Hermite curves as: • The