CS 445 / 645Introduction to Computer Graphics

Lecture 23Lecture 23

BBézier Curvesézier Curves

Lecture 23Lecture 23

BBézier Curvesézier Curves

Assign 4

Task ListTask List

• Get fluid to give you four viewing windowsGet fluid to give you four viewing windows

• Build an interface so you can click down a series of pointsBuild an interface so you can click down a series of points

– Render line segments between pointsRender line segments between points

– Design a way to insert, move, and delete these pointsDesign a way to insert, move, and delete these points

How do you know if you are adding a point to the front How do you know if you are adding a point to the front of the spline versus adding a point after the first point of the spline versus adding a point after the first point of the spline?of the spline?

Task ListTask List

• Get fluid to give you four viewing windowsGet fluid to give you four viewing windows

• Build an interface so you can click down a series of pointsBuild an interface so you can click down a series of points

– Render line segments between pointsRender line segments between points

– Design a way to insert, move, and delete these pointsDesign a way to insert, move, and delete these points

How do you know if you are adding a point to the front How do you know if you are adding a point to the front of the spline versus adding a point after the first point of the spline versus adding a point after the first point of the spline?of the spline?

Assign 4

Task ListTask List

• Build a camera model for each windowBuild a camera model for each window

– Error in assignment: Orthographic cameras don’t zoom. They Error in assignment: Orthographic cameras don’t zoom. They change their viewing frusta (left, right, top, bottom)change their viewing frusta (left, right, top, bottom)

• Perspective camera needs track ball (Angel 4.10.2, A.11)Perspective camera needs track ball (Angel 4.10.2, A.11)

• First-person camera that slides along curveFirst-person camera that slides along curve

– acceptable to have uniform increments of tacceptable to have uniform increments of t

Even though this may lead to non-uniform velocitiesEven though this may lead to non-uniform velocities

– Curve derivative defines look-at direction, but what about V-up?Curve derivative defines look-at direction, but what about V-up?

Task ListTask List

• Build a camera model for each windowBuild a camera model for each window

– Error in assignment: Orthographic cameras don’t zoom. They Error in assignment: Orthographic cameras don’t zoom. They change their viewing frusta (left, right, top, bottom)change their viewing frusta (left, right, top, bottom)

• Perspective camera needs track ball (Angel 4.10.2, A.11)Perspective camera needs track ball (Angel 4.10.2, A.11)

• First-person camera that slides along curveFirst-person camera that slides along curve

– acceptable to have uniform increments of tacceptable to have uniform increments of t

Even though this may lead to non-uniform velocitiesEven though this may lead to non-uniform velocities

– Curve derivative defines look-at direction, but what about V-up?Curve derivative defines look-at direction, but what about V-up?

Assign 4

Task ListTask List

• Build three B-Splines (x, y, and z) from vertex listBuild three B-Splines (x, y, and z) from vertex list

• Convert B-Spline into Bezier SplinesConvert B-Spline into Bezier Splines

• Recursively subdivide Bezier Spline (Angel 10.9.2)Recursively subdivide Bezier Spline (Angel 10.9.2)

– Render subdivided Bezier Spline as line segmentsRender subdivided Bezier Spline as line segments

• Incorporate Yannick’s pipe code to recreate a first-person Incorporate Yannick’s pipe code to recreate a first-person view of riding through pipeview of riding through pipe

Task ListTask List

• Build three B-Splines (x, y, and z) from vertex listBuild three B-Splines (x, y, and z) from vertex list

• Convert B-Spline into Bezier SplinesConvert B-Spline into Bezier Splines

• Recursively subdivide Bezier Spline (Angel 10.9.2)Recursively subdivide Bezier Spline (Angel 10.9.2)

– Render subdivided Bezier Spline as line segmentsRender subdivided Bezier Spline as line segments

• Incorporate Yannick’s pipe code to recreate a first-person Incorporate Yannick’s pipe code to recreate a first-person view of riding through pipeview of riding through pipe

Virtual Trackball

Can we use the mouse to control the 2-D rotation Can we use the mouse to control the 2-D rotation of a viewing volume?of a viewing volume?

Imagine a track ball (draw it on board)Imagine a track ball (draw it on board)

• User moves point on ball from (x, y, z) to (a, b, c)User moves point on ball from (x, y, z) to (a, b, c)

Imagine the points projected onto the groundImagine the points projected onto the ground

• User moves point on ground from (x, 0, z) to (a, 0, c)User moves point on ground from (x, 0, z) to (a, 0, c)

Can we use the mouse to control the 2-D rotation Can we use the mouse to control the 2-D rotation of a viewing volume?of a viewing volume?

Imagine a track ball (draw it on board)Imagine a track ball (draw it on board)

• User moves point on ball from (x, y, z) to (a, b, c)User moves point on ball from (x, y, z) to (a, b, c)

Imagine the points projected onto the groundImagine the points projected onto the ground

• User moves point on ground from (x, 0, z) to (a, 0, c)User moves point on ground from (x, 0, z) to (a, 0, c)

Trackball

Movement of points on track ball can be inferred Movement of points on track ball can be inferred from mouse drags on screenfrom mouse drags on screen

Inverse problemInverse problem

• Where on trackball does (a, 0, c) hit?Where on trackball does (a, 0, c) hit?

• Ball is unit sphere, so ||x, y, z|| = 1.0Ball is unit sphere, so ||x, y, z|| = 1.0

• x = a, z = c, y = solve for itx = a, z = c, y = solve for it

Movement of points on track ball can be inferred Movement of points on track ball can be inferred from mouse drags on screenfrom mouse drags on screen

Inverse problemInverse problem

• Where on trackball does (a, 0, c) hit?Where on trackball does (a, 0, c) hit?

• Ball is unit sphere, so ||x, y, z|| = 1.0Ball is unit sphere, so ||x, y, z|| = 1.0

• x = a, z = c, y = solve for itx = a, z = c, y = solve for it

TrackballUser defines two pointsUser defines two points

• Place where first clicked X = (x, y, z)Place where first clicked X = (x, y, z)

• Place where released A = (a, b, c)Place where released A = (a, b, c)

Ball rotates along axis perp to line defined by these two pointsBall rotates along axis perp to line defined by these two points

• compute cross produce of lines to origin: (X – O) x (A – O)compute cross produce of lines to origin: (X – O) x (A – O)

Ball rotates by amount proportional to distance between linesBall rotates by amount proportional to distance between lines

• magnitude of cross product tells us angle between lines (dot product too)magnitude of cross product tells us angle between lines (dot product too)

• |sin |sin | = ||cross product||| = ||cross product||

Compute rotation matrix and use it to rotate worldCompute rotation matrix and use it to rotate world

User defines two pointsUser defines two points

• Place where first clicked X = (x, y, z)Place where first clicked X = (x, y, z)

• Place where released A = (a, b, c)Place where released A = (a, b, c)

Ball rotates along axis perp to line defined by these two pointsBall rotates along axis perp to line defined by these two points

• compute cross produce of lines to origin: (X – O) x (A – O)compute cross produce of lines to origin: (X – O) x (A – O)

Ball rotates by amount proportional to distance between linesBall rotates by amount proportional to distance between lines

• magnitude of cross product tells us angle between lines (dot product too)magnitude of cross product tells us angle between lines (dot product too)

• |sin |sin | = ||cross product||| = ||cross product||

Compute rotation matrix and use it to rotate worldCompute rotation matrix and use it to rotate world

Splines - History

Draftsman use ‘ducks’ and Draftsman use ‘ducks’ and strips of wood (splines) to strips of wood (splines) to draw curvesdraw curves

Wood splines have second-Wood splines have second-order continuityorder continuity

And pass through the And pass through the control pointscontrol points

Draftsman use ‘ducks’ and Draftsman use ‘ducks’ and strips of wood (splines) to strips of wood (splines) to draw curvesdraw curves

Wood splines have second-Wood splines have second-order continuityorder continuity

And pass through the And pass through the control pointscontrol points

A Duck (weight)

Ducks trace out curve

Bézier Curves

Similar to Hermite, but more intuitive definition of Similar to Hermite, but more intuitive definition of endpoint derivativesendpoint derivatives

Four control points, two of which are knotsFour control points, two of which are knots

Similar to Hermite, but more intuitive definition of Similar to Hermite, but more intuitive definition of endpoint derivativesendpoint derivatives

Four control points, two of which are knotsFour control points, two of which are knots

Bézier Curves

The derivative values of the Bezier Curve at the The derivative values of the Bezier Curve at the knots are dependent on the adjacent pointsknots are dependent on the adjacent points

The scalar 3 was selected just for this curve The scalar 3 was selected just for this curve

The derivative values of the Bezier Curve at the The derivative values of the Bezier Curve at the knots are dependent on the adjacent pointsknots are dependent on the adjacent points

The scalar 3 was selected just for this curve The scalar 3 was selected just for this curve

Bézier vs. Hermite

We can write our Bezier in terms of HermiteWe can write our Bezier in terms of Hermite

• Note this is just matrix form of previous equationsNote this is just matrix form of previous equations

We can write our Bezier in terms of HermiteWe can write our Bezier in terms of Hermite

• Note this is just matrix form of previous equationsNote this is just matrix form of previous equations

Bézier vs. Hermite

Now substitute this in for previous HermiteNow substitute this in for previous HermiteNow substitute this in for previous HermiteNow substitute this in for previous Hermite

Bézier Basis and Geometry Matrices

Matrix FormMatrix Form

But why is MBut why is MBezierBezier a good basis matrix? a good basis matrix?

Matrix FormMatrix Form

But why is MBut why is MBezierBezier a good basis matrix? a good basis matrix?

Bézier Blending Functions

Look at the blending Look at the blending functionsfunctions

This family of This family of polynomials is calledpolynomials is calledorder-3 Bernstein order-3 Bernstein PolynomialsPolynomials• C(3, k) tC(3, k) tkk (1-t) (1-t)3-k3-k; 0<= k <= 3; 0<= k <= 3

• They are all positive in interval [0,1]They are all positive in interval [0,1]

• Their sum is equal to 1Their sum is equal to 1

Look at the blending Look at the blending functionsfunctions

This family of This family of polynomials is calledpolynomials is calledorder-3 Bernstein order-3 Bernstein PolynomialsPolynomials• C(3, k) tC(3, k) tkk (1-t) (1-t)3-k3-k; 0<= k <= 3; 0<= k <= 3

• They are all positive in interval [0,1]They are all positive in interval [0,1]

• Their sum is equal to 1Their sum is equal to 1

Bézier Blending Functions

Thus, every point on curve is Thus, every point on curve is linear combination of the linear combination of the control pointscontrol points

The weights of the The weights of the combination are all positivecombination are all positive

The sum of the weights is 1The sum of the weights is 1

Therefore, the curve is a Therefore, the curve is a convex combination of the convex combination of the control pointscontrol points

Thus, every point on curve is Thus, every point on curve is linear combination of the linear combination of the control pointscontrol points

The weights of the The weights of the combination are all positivecombination are all positive

The sum of the weights is 1The sum of the weights is 1

Therefore, the curve is a Therefore, the curve is a convex combination of the convex combination of the control pointscontrol points

Bézier Curves

Will always remain within bounding region Will always remain within bounding region defined by control pointsdefined by control points

Will always remain within bounding region Will always remain within bounding region defined by control pointsdefined by control points

Bézier Curves

BezierBezierBezierBezier

Why more spline slides?

Bezier and Hermite splines have global influenceBezier and Hermite splines have global influence

• Piecewise Bezier or Hermite don’t enforce derivative continuity at join Piecewise Bezier or Hermite don’t enforce derivative continuity at join pointspoints

• Moving one control point affects the entire curveMoving one control point affects the entire curve

B-splinesB-splines consist of curve segments whose polynomial consist of curve segments whose polynomial coefficients depend on just a few control pointscoefficients depend on just a few control points

• Local controlLocal control

Examples of SplinesExamples of Splines

Bezier and Hermite splines have global influenceBezier and Hermite splines have global influence

• Piecewise Bezier or Hermite don’t enforce derivative continuity at join Piecewise Bezier or Hermite don’t enforce derivative continuity at join pointspoints

• Moving one control point affects the entire curveMoving one control point affects the entire curve

B-splinesB-splines consist of curve segments whose polynomial consist of curve segments whose polynomial coefficients depend on just a few control pointscoefficients depend on just a few control points

• Local controlLocal control

Examples of SplinesExamples of Splines

B-Spline Curve

Start with a sequence of control pointsStart with a sequence of control points

Select four from middle of sequence Select four from middle of sequence (p(pi-2i-2, p, pi-1i-1, p, pii, p, pi+1i+1) ) dd

• Bezier and Hermite goes between pBezier and Hermite goes between p i-2i-2 and p and pi+1i+1

• B-Spline doesn’t interpolate (touch) any of them but B-Spline doesn’t interpolate (touch) any of them but approximates the going through papproximates the going through p i-1i-1 and p and pii

Start with a sequence of control pointsStart with a sequence of control points

Select four from middle of sequence Select four from middle of sequence (p(pi-2i-2, p, pi-1i-1, p, pii, p, pi+1i+1) ) dd

• Bezier and Hermite goes between pBezier and Hermite goes between p i-2i-2 and p and pi+1i+1

• B-Spline doesn’t interpolate (touch) any of them but B-Spline doesn’t interpolate (touch) any of them but approximates the going through papproximates the going through p i-1i-1 and p and pii

pp00pp00 pp44pp44

pp22pp22pp11pp11

pp33pp33

pp55pp55

pp66pp66

QQ33QQ33

QQ44QQ44

QQ55QQ55

QQ66QQ66

Uniform B-Splines

ApproximatingApproximating Splines Splines

Approximates n+1 control pointsApproximates n+1 control points

• PP00, P, P11, …, P, …, Pnn, n , n ¸̧ 3 3

Curve consists of n –2 cubic polynomial segmentsCurve consists of n –2 cubic polynomial segments

• QQ33, Q, Q44, … Q, … Qnn

t varies along B-spline as Qt varies along B-spline as Qii: t: tii <= t < t <= t < ti+1i+1

ttii (i = integer) are (i = integer) are knot pointsknot points that join segment Q that join segment Qi-1i-1 to Q to Qii

Curve is Curve is uniformuniform because knots are spaced at equal intervals of because knots are spaced at equal intervals of parameter,parameter, tt

ApproximatingApproximating Splines Splines

Approximates n+1 control pointsApproximates n+1 control points

• PP00, P, P11, …, P, …, Pnn, n , n ¸̧ 3 3

Curve consists of n –2 cubic polynomial segmentsCurve consists of n –2 cubic polynomial segments

• QQ33, Q, Q44, … Q, … Qnn

t varies along B-spline as Qt varies along B-spline as Qii: t: tii <= t < t <= t < ti+1i+1

ttii (i = integer) are (i = integer) are knot pointsknot points that join segment Q that join segment Qi-1i-1 to Q to Qii

Curve is Curve is uniformuniform because knots are spaced at equal intervals of because knots are spaced at equal intervals of parameter,parameter, tt

Uniform B-Splines

First curve segment, QFirst curve segment, Q33, is defined by first four , is defined by first four

control pointscontrol points

Last curve segment, QLast curve segment, Qmm, is defined by last four , is defined by last four

control points, Pcontrol points, Pm-3m-3, P, Pm-2m-2, P, Pm-1m-1, P, Pmm

Each control point affects four curve segmentsEach control point affects four curve segments

First curve segment, QFirst curve segment, Q33, is defined by first four , is defined by first four

control pointscontrol points

Last curve segment, QLast curve segment, Qmm, is defined by last four , is defined by last four

control points, Pcontrol points, Pm-3m-3, P, Pm-2m-2, P, Pm-1m-1, P, Pmm

Each control point affects four curve segmentsEach control point affects four curve segments

B-spline Basis Matrix

Formulate 16 equations to solve the 16 unknownsFormulate 16 equations to solve the 16 unknowns

The 16 equations enforce the CThe 16 equations enforce the C00, C, C11, and C, and C22

continuity between adjoining segments, Qcontinuity between adjoining segments, Q

Formulate 16 equations to solve the 16 unknownsFormulate 16 equations to solve the 16 unknowns

The 16 equations enforce the CThe 16 equations enforce the C00, C, C11, and C, and C22

continuity between adjoining segments, Qcontinuity between adjoining segments, Q

0141

0303

0363

1331

6

1splineBM

B-Spline Basis Matrix

Note the order of the rows in my MNote the order of the rows in my MB-Spline B-Spline is is

different from in the bookdifferent from in the book

• Observe also that the order in which I number the points is Observe also that the order in which I number the points is differentdifferent

• Therefore my matrix aligns with the book’s matrix if you Therefore my matrix aligns with the book’s matrix if you reorder the points, and thus reorder the rows of the matrixreorder the points, and thus reorder the rows of the matrix

Note the order of the rows in my MNote the order of the rows in my MB-Spline B-Spline is is

different from in the bookdifferent from in the book

• Observe also that the order in which I number the points is Observe also that the order in which I number the points is differentdifferent

• Therefore my matrix aligns with the book’s matrix if you Therefore my matrix aligns with the book’s matrix if you reorder the points, and thus reorder the rows of the matrixreorder the points, and thus reorder the rows of the matrix

B-Spline

Points along B-Spline are computed just as with Points along B-Spline are computed just as with Bezier CurvesBezier Curves

Points along B-Spline are computed just as with Points along B-Spline are computed just as with Bezier CurvesBezier Curves

PUMtQ SplineBi

3

2

123

0141

0303

0363

1331

6

11

i

i

i

i

i

p

p

p

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B-Spline

By far the most popular spline usedBy far the most popular spline used

CC00, C, C11, and C, and C22 continuous continuous

By far the most popular spline usedBy far the most popular spline used

CC00, C, C11, and C, and C22 continuous continuous

B-Spline

Locality of pointsLocality of pointsLocality of pointsLocality of points

Nonuniform, Rational B-Splines(NURBS)

The native geometry element in MayaThe native geometry element in Maya

Models are composed of surfaces defined by Models are composed of surfaces defined by NURBS, not polygonsNURBS, not polygons

NURBS are smoothNURBS are smooth

NURBS require effort to make non-smoothNURBS require effort to make non-smooth

The native geometry element in MayaThe native geometry element in Maya

Models are composed of surfaces defined by Models are composed of surfaces defined by NURBS, not polygonsNURBS, not polygons

NURBS are smoothNURBS are smooth

NURBS require effort to make non-smoothNURBS require effort to make non-smooth

NURBs

To generate three-dimension pointsTo generate three-dimension points

• Four B-Splines are usedFour B-Splines are used

– Three define the x, y, and z positionThree define the x, y, and z position

– One defines a weighting termOne defines a weighting term

• The 3-D point output by a NURB is the x, y, z coordinates The 3-D point output by a NURB is the x, y, z coordinates divided by the weighting term (like homogeneous divided by the weighting term (like homogeneous coordinates)coordinates)

To generate three-dimension pointsTo generate three-dimension points

• Four B-Splines are usedFour B-Splines are used

– Three define the x, y, and z positionThree define the x, y, and z position

– One defines a weighting termOne defines a weighting term

• The 3-D point output by a NURB is the x, y, z coordinates The 3-D point output by a NURB is the x, y, z coordinates divided by the weighting term (like homogeneous divided by the weighting term (like homogeneous coordinates)coordinates)

What is a NURB?

Nonuniform:Nonuniform: The amount of parameter, t, that is The amount of parameter, t, that is used to model each curve segment varies used to model each curve segment varies • Nonuniformity permits either CNonuniformity permits either C22, C, C11, or C, or C00 continuity at continuity at

join points between curve segmentsjoin points between curve segments

• Nonuniformity permits control points to be added to Nonuniformity permits control points to be added to middle of curvemiddle of curve

Nonuniform:Nonuniform: The amount of parameter, t, that is The amount of parameter, t, that is used to model each curve segment varies used to model each curve segment varies • Nonuniformity permits either CNonuniformity permits either C22, C, C11, or C, or C00 continuity at continuity at

join points between curve segmentsjoin points between curve segments

• Nonuniformity permits control points to be added to Nonuniformity permits control points to be added to middle of curvemiddle of curve

What do we get?

NURBs are invariant under rotation, scaling, NURBs are invariant under rotation, scaling, translation, and perspective transformations of translation, and perspective transformations of the control points (nonrational curves are not the control points (nonrational curves are not preserved under perspective projection)preserved under perspective projection)• This means you can transform the control points and redraw the This means you can transform the control points and redraw the

curve using the transformed pointscurve using the transformed points

• If this weren’t true you’d have to sample curve to many points If this weren’t true you’d have to sample curve to many points and transform each point individuallyand transform each point individually

• B-spline is preserved under affine transformations, but that is allB-spline is preserved under affine transformations, but that is all

NURBs are invariant under rotation, scaling, NURBs are invariant under rotation, scaling, translation, and perspective transformations of translation, and perspective transformations of the control points (nonrational curves are not the control points (nonrational curves are not preserved under perspective projection)preserved under perspective projection)• This means you can transform the control points and redraw the This means you can transform the control points and redraw the

curve using the transformed pointscurve using the transformed points

• If this weren’t true you’d have to sample curve to many points If this weren’t true you’d have to sample curve to many points and transform each point individuallyand transform each point individually

• B-spline is preserved under affine transformations, but that is allB-spline is preserved under affine transformations, but that is all

Converting Between Splines

Consider two spline basis formulations for two Consider two spline basis formulations for two spline typesspline types

Consider two spline basis formulations for two Consider two spline basis formulations for two spline typesspline types

Converting Between Splines

We can transform the control points from one We can transform the control points from one spline basis to anotherspline basis to another

We can transform the control points from one We can transform the control points from one spline basis to anotherspline basis to another

Converting Between Splines

With this conversion, we can convert a B-Spline With this conversion, we can convert a B-Spline into a Bezier Splineinto a Bezier Spline

Bezier Splines are easy to renderBezier Splines are easy to render

With this conversion, we can convert a B-Spline With this conversion, we can convert a B-Spline into a Bezier Splineinto a Bezier Spline

Bezier Splines are easy to renderBezier Splines are easy to render

Rendering Splines

Horner’s MethodHorner’s Method

Incremental (Forward Difference) MethodIncremental (Forward Difference) Method

Subdivision MethodsSubdivision Methods

Horner’s MethodHorner’s Method

Incremental (Forward Difference) MethodIncremental (Forward Difference) Method

Subdivision MethodsSubdivision Methods

Horner’s Method

Three multiplicationsThree multiplications

Three additionsThree additions

Three multiplicationsThree multiplications

Three additionsThree additions

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Forward Difference

But this still is expensive to computeBut this still is expensive to compute

• Solve for change at k (Solve for change at k (kk) and change at k+1 () and change at k+1 (k+1k+1))

• Boot strap with initial values for xBoot strap with initial values for x00, , 00, and , and 11

• Compute xCompute x33 by adding x by adding x00 + + 00 + + 11

But this still is expensive to computeBut this still is expensive to compute

• Solve for change at k (Solve for change at k (kk) and change at k+1 () and change at k+1 (k+1k+1))

• Boot strap with initial values for xBoot strap with initial values for x00, , 00, and , and 11

• Compute xCompute x33 by adding x by adding x00 + + 00 + + 11

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231

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Subdivision Methods

BezierBezierBezierBezier

Rendering Bezier Spline

public void spline(ControlPoint p0, ControlPoint p1, ControlPoint p2, ControlPoint p3, int pix) { float len = ControlPoint.dist(p0,p1) + ControlPoint.dist(p1,p2) + ControlPoint.dist(p2,p3); float chord = ControlPoint.dist(p0,p3); if (Math.abs(len - chord) < 0.25f) return; fatPixel(pix, p0.x, p0.y); ControlPoint p11 = ControlPoint.midpoint(p0, p1); ControlPoint tmp = ControlPoint.midpoint(p1, p2); ControlPoint p12 = ControlPoint.midpoint(p11, tmp); ControlPoint p22 = ControlPoint.midpoint(p2, p3); ControlPoint p21 = ControlPoint.midpoint(p22, tmp); ControlPoint p20 = ControlPoint.midpoint(p12, p21); spline(p20, p12, p11, p0, pix); spline(p3, p22, p21, p20, pix); }