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• pramook_at_gmail.com

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CS 445 / 645Introduction to Computer Graphics

• Lecture 22
• Hermite Splines

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Splines Old School
Duck
Spline
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Representations of Curves

• Use a sequence of points
• Piecewise linear - does not accurately model a
  smooth line
• Tedious to create list of points
• Expensive to manipulate curve because all points
  must be repositioned
• Instead, model curve as piecewise-polynomial
• x x(t), y y(t), z z(t)
• where x(), y(), z() are polynomials

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Specifying Curves (hyperlink)

• Control Points
• A set of points that influence the curves shape
• Knots
• Control points that lie on the curve
• Interpolating Splines
• Curves that pass through the control points
  (knots)
• Approximating Splines
• Control points merely influence shape

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Parametric Curves

• Very flexible representation
• They are not required to be functions
• They can be multivalued with respect to any
  dimension

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Cubic Polynomials

• x(t) axt3 bxt2 cxt dx
• Similarly for y(t) and z(t)
• Let t (0 lt t lt 1)
• Let T t3 t2 t 1
• Coefficient Matrix C
• Curve Q(t) TC

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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

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Parametric Continuity

• C1 is tangent continuity (velocity)
• C2 is 2nd derivative continuity (acceleration)
• Matching direction and magnitude of dn / dtn
• Cn continous

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Geometric Continuity

• 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

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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?

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Parametric Cubic Splines

• Can represent this as a matrix too

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Coefficients

• So how do we select the coefficients?
• ax bx cx dx and ay by cy dy must satisfy the
  constraints defined by the knots and the
  continuity conditions

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Parametric Curves

• Difficult to conceptualize curve as x(t) axt3
  bxt2 cxt dx(artists dont think in terms
  of coefficients of cubics)
• Instead, define curve as weighted combination of
  4 well-defined cubic polynomials(wait a second!
  Artists dont think this way either!)
• Each curve type defines different cubic
  polynomials and weighting schemes

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Parametric Curves

• Hermite two endpoints and two endpoint tangent
  vectors
• Bezier - two endpoints and two other points that
  define the endpoint tangent vectors
• Splines four control points
• C1 and C2 continuity at the join points
• Come close to their control points, but not
  guaranteed to touch them
• Examples of Splines

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Hermite Cubic Splines

• An example of knot and continuity constraints

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Hermite Cubic Splines

• One cubic curve for each dimension
• A curve constrained to x/y-plane has two curves

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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, p1
• (x, y) position at t 1, p2
• (x, y) derivative at t 0, dp/dt
• (x, y) derivative at t 1, dp/dt

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Hermite Cubic Spline

• We know
• (x, y) position at t 0, p1

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Hermite Cubic Spline

• We know
• (x, y) position at t 1, p2

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Hermite Cubic Splines

• So far we have four equations, but we have eight
  unknowns
• Use the derivatives

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Hermite Cubic Spline

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

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Hermite Cubic Spline

• We know
• (x, y) derivative at t 1, dp/dt

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Hermite Specification

• Matrix equation for Hermite Curve

t3 t2 t1 t0
t 0
p1
t 1
p2
t 0
r p1
r p2
t 1
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Solve Hermite Matrix
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Spline and Geometry Matrices
MHermite
GHermite
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Resulting Hermite Spline Equation
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Sample Hermite Curves
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Blending Functions

• By multiplying first two matrices in lower-left
  equation, you have four functions of t that
  blend the four control parameters
• These are blendingfunctions

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Hermite Blending Functions

• If you plot the blending functions on the
  parameter t

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Hermite Blending Functions

• Remember, eachblending functionreflects
  influenceof P1, P2, DP1, DP2on splines shape

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CS 445 / 645Introduction to Computer Graphics

• Lecture 23
• Bézier Curves

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Splines - History

• Draftsman use ducks and strips of wood
  (splines) to draw curves
• Wood splines have second-order continuity
• And pass through the control points

A Duck (weight)
Ducks trace out curve
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Bézier Curves

• Similar to Hermite, but more intuitive definition
  of endpoint derivatives
• Four control points, two of which are knots

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Bézier Curves

• The derivative values of the Bezier Curve at the
  knots are dependent on the adjacent points
• The scalar 3 was selected just for this curve

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Bézier vs. Hermite

• We can write our Bezier in terms of Hermite
• Note this is just matrix form of previous
  equations

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Bézier vs. Hermite

• Now substitute this in for previous Hermite

MBezier
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Bézier Basis and Geometry Matrices

• Matrix Form
• But why is MBezier a good basis matrix?

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Bézier Blending Functions

• Look at the blending functions
• This family of polynomials is calledorder-3
  Bernstein Polynomials
• C(3, k) tk (1-t)3-k 0lt k lt 3
• They are all positive in interval 0,1
• Their sum is equal to 1

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Bézier Blending Functions

• Thus, every point on curve is linear combination
  of the control points
• The weights of the combination are all positive
• The sum of the weights is 1
• Therefore, the curve is a convex combination of
  the control points

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Convex combination of control points

• Will always remain within bounding region (convex
  hull) defined by control points

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Why more spline slides?

• Bezier and Hermite splines have global influence
• One could create a Bezier curve that required 15
  points to define the curve
• Moving any one control point would affect the
  entire curve
• Piecewise Bezier or Hermite dont suffer from
  this, but they dont enforce derivative
  continuity at join points
• B-splines consist of curve segments whose
  polynomial coefficients depend on just a few
  control points
• Local control
• Examples of Splines

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B-Spline Curve (cubic periodic)

• Start with a sequence of control points
• Select four from middle of sequence (pi-2, pi-1,
  pi, pi1) d
• Bezier and Hermite goes between pi-2 and pi1
• B-Spline doesnt interpolate (touch) any of them
  but approximates going through pi-1 and pi

p2
p6
p1
Q4
Q5
t4
t5
Q3
Q6
t6
p3
t3
t7
p0
p4
p5
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Uniform B-Splines

• Approximating Splines
• Approximates n1 control points
• P0, P1, , Pn, n 3
• Curve consists of n 2 cubic polynomial segments
• Q3, Q4, Qn
• t varies along B-spline as Qi ti lt t lt ti1
• ti (i integer) are knot points that join
  segment Qi to Qi1
• Curve is uniform because knots are spaced at
  equal intervals of parameter, t

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Uniform B-Splines

• First curve segment, Q3, is defined by first four
  control points
• Last curve segment, Qm, is defined by last four
  control points, Pm-3, Pm-2, Pm-1, Pm
• Each control point affects four curve segments

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B-spline Basis Matrix

• Formulate 16 equations to solve the 16 unknowns
• The 16 equations enforce the C0, C1, and C2
  continuity between adjoining segments, Q

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B-Spline

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

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B-Spline

• By far the most popular spline used
• C0, C1, and C2 continuous

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Nonuniform, Rational B-Splines(NURBS)

• The native geometry element in Maya
• Models are composed of surfaces defined by NURBS,
  not polygons
• NURBS are smooth
• NURBS require effort to make non-smooth

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Converting Between Splines

• Consider two spline basis formulations for two
  spline types

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Converting Between Splines

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

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Converting Between Splines

• With this conversion, we can convert a B-Spline
  into a Bezier Spline
• Bezier Splines are easy to render

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Rendering Splines

• Horners Method
• Incremental (Forward Difference) Method
• Subdivision Methods

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Horners Method

• Three multiplications
• Three additions

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Forward Difference

• But this still is expensive to compute
• Solve for change at k (Dk) and change at k1
  (Dk1)
• Boot strap with initial values for x0, D0, and D1
• Compute x3 by adding x0 D0 D1

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Subdivision Methods

• Bezier

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Rendering Bezier Spline
public void spline(ControlPoint p0,
ControlPoint p1,
ControlPoint p2, ControlPoint p3, int pix)
float len ControlPoint.dist(p0,p
1) ControlPoint.dist(p1,p2)
ControlPoint.dist(p2,p3)
float chord ControlPoint.dist(p0,p3)
if (Math.abs(len - chord) lt 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)
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???????????

• ??????????????????????????????? ? ?????? (x,y,z)
  ???
• ???????? (x,y,z) ??????????????????????????

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????????

• ??????????????????????????????????? 60 ??????????
  (1,1,1)
• ???????????????????????????
• ???????? cos ??? sin
• ?????????????????????????

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• ?????????
• ??????
• ????????????????

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?????????????????

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• ??????????????????????????????????????????????????
  ?

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????????

• ???
• ???????

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????????

• q1 ????????????????? 90 ???? ?????? (3/5, 0, 4/5)
• q2 ????????????????? 60 ???? ?????? (3/5, 0, 4/5)
• ?????? q1q2 ????????????????? 60 ???????????????
  90 ????
• ?????????????????? 150 ?????????? (3/5, 0, 4/5)
• ??????

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Slerp

• ????????? slerp ?????????????
• ?????????????????? slerp(q0,q1,?) ????????? ?
  ????????????????????? ??? 0 ??? 1 ?????? plot
  quaternion ???????? ???????????
  ????????????????????????????????? geodesic
  ??????????????????? 4 ???????????????????????? q0
  ??? q1
• ??? ? ??????????????????????? geodesic ???
  ????????
• ??? ? 0 ????????? q0
• ??? ? 1 ????????? q1
• ??? ? 0.5 ???????????????????? q0 ??? q1 ????
• ???

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Slerp
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????????

• ???
• ???????

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????????

• ????????? x component ??? z component ???? 0
• ????????????????? x ??? z ????????????????? 0
  ???? ????????????? geodesic ?????????????????? x
  ??? z ??????? 0 (??????????????????????)
• ?????????????????????????? geodesic
  ??????????????????????? 2 ???? ???????????????????
  ??????????????? w ?????? y

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????????

• ?????????? q0 ??? q1 ??? 90 ????
• slerp(q0,q1,1/3) ????????????????????? q0 ????
  1/3 ?????????? 90 ???? ????????????? 30 ???????
  q0
• ?????? slerp(q0,q1,1/3) ?????????? (w,y) ???????
• ????????

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????????
y
q1
2/3
slerp(q0,q1,1/3)
1/3
w
q0