Geometric Modeling91.580.201

CurvesMortenson Chapter 2-5

and Angel Chapter 9

Curve Basics• Curve: Locus of a point moving with 1 degree of freedom.• Some types of equations to describe curves:

– Intrinsic• No reliance on external frame of reference

– Explicit• Value of dependent variable in terms of independent variable(s)• e.g. y = f (x)

– Implicit• e.g. f (x,y) = 0

– Parametric• Express value of each spatial variable in terms of an independent

variable (the parameter)• e.g. for parameter u in 3D:

x = x (u)y = y (u)z = z (u)

source: Mortenson, source: Mortenson, AngelAngel

Intrinsic Definition• No reliance on external frame of reference• Requires 2 equations as functions of arc

length* s:1) Curvature:

2) Torsion:

• For plane curves, alternatively:

)(1

sf

source: Mortensonsource: Mortenson

)(sg

*length measured along the curve

Torsion (in 3D) measures how much curve deviates from a plane curve.

Treated in more detail in Chapter 12 of Mortenson and Chapters 10, 19 of Farin.

ds

d

1

Explicit Form

• Value of dependent variable in terms of independent variable(s)– e.g. in 2D: one y value for each x value: y = f (x)– e.g. in 3D: y = f (x) and z = g (x)

• Axis-dependent• Not guaranteed to exist

– e.g. for 2D circle only one half is described by:

• Special cases can be problematic– e.g. y = mx + h is inappropriate for vertical lines

22 xry

source: Mortenson, source: Mortenson, AngelAngel

Implicit Form

• General form: f (x,y) = 0• Examples:

– Straight line: Ax + By + C=0 – Conic curve: Ax2 + 2Bxy + Cy2 + Dx + Ey + F=0

• Axis-dependent• Can represent all lines and circles and

more…• Some implicit forms are hard to

parameterize.

source: Mortenson, source: Mortenson, AngelAngel

Parametric Form

• Express value of each spatial variable in terms of an independent variable (the parameter)– e.g. for parameter u in 3D:

x = x (u)y = y (u)z = z (u)

• For a curve segment, typically • Curve segments can be joined to

form composite spline* (bendable) curves.

• Parametric form of a given curve is not necessarily unique.

• Every parametric form has an implicit form.

]1,0[u

source: Mortenson, source: Mortenson, AngelAngel

*see Mortenson p. 105 (or Farin p. 167) for spline definition as a “minimum energy curve.”

Parametric Form

source: Mortensonsource: Mortenson

vector

tangent vector*

*unless otherwise noted, tangent vectors not drawn to scale.

du

udzz

du

udyy

du

udxx uuu )(

)(

)(

Vector components of pu(u):

Parametric Form Advantages

• Allow separation of variables and direct computation of point coordinates.

• Accommodate all slopes.• Bounded when parameter lies in a

bounded interval.• More degrees of freedom to control curve

shape.• Various forms of continuity can be

enforced at curve segment join points.

source: Mortensonsource: MortensonSee Mortenson p.46 for more…

Continuity at Join Points• Discontinuous: physical separation• Parametric Continuity

• Positional (C0 ): no physical separation• C1 : C0 and matching first derivatives• C2 : C1 and matching second derivatives

• Geometric Continuity• Positional (G0 ) = C0

• Tangential (G1) : G0 and tangents are proportional, point in same direction, but magnitudes may differ

• Curvature (G2) : G1 and tangent lengths are the same and rate of length change is the same

source: Mortenson, Angel, source: Mortenson, Angel, WikiWiki

4 Typical Types of Parametric Curves

• Interpolating– Curve passes through all control points.

• Hermite– Defined by its 2 endpoints and tangent vectors at endpoints.– Interpolates all its control points.– Not invariant under affine transformations.– Special case of Bezier and B-Spline.

• Bezier– Interpolates first and last control points.– Curve is tangent to first and last segments of control polygon.– Easy to subdivide.– Curve segment lies within convex hull of control polygon.– Variation-diminishing.– Special case of B-spline.

• B-Spline– Not guaranteed to interpolate control points.– Invariant under affine transformations.– Curve segment lies within convex hull of control polygon.– Variation-diminishing.– Greater local control than Bezier.

Control points influence curve shape.

source: Mortenson, source: Mortenson, AngelAngel

Interpolating

• Interpolates all control points.• Geometric form:

• Rarely used due to lack of derivative continuity at curve segment join points.

source: Angelsource: Angel

n

iii ubu

0

)()( pp

cubic case with equally spaced parameter values

Hermite

• Geometric form (cubic case):

• Hermite curves can provide C1 continuity at curve segment join points.

234

233

232

231

)(

2)(

32)(

132)(

uuuF

uuuuF

uuuF

uuuF

source: Mortensonsource: Mortenson

14031201 )()()()()( uu uFuFuFuFu ppppp

Bezier• Geometric form (cubic case):

• Bezier curves can provide C1 continuity at curve segment join points.

n

iini uBu

0, )()( pp

source: Mortensonsource: Mortenson

inini uu

i

nuB

)1()(,

Bernstein polynomials.

n+1 = number of control points = degree + 1

Adding a control point elevates degree by 1.

Convex combination, so Bezier curve points all lie within convex hull of control polygon.

n

ini uB

0, 1)(

Rational form is invariant under perspective transformation:where hi are projective space coordinates (weights)

n

inii

n

iinii

uBh

uBhu

0,

0,

)(

)()(

pp

B-Spline

n

iiKi uNu

0, )()( pp

otherwise 0)(

if 1)(

1,

11,

uN

tutuN

i

iii

• Geometric form (non-uniform, non-rational case), where K controls degree (K -1) of basis functions:

• Cubic B-splines can provide C2 continuity at curve segment join points.

Convex combination, so B-spline curve points all lie within convex hull of control polygon.

n

iKi uN

0, 1)(

Rational form (NURBS) is invariant under perspective transformation, where hi are projective space coordinates (weights).

n

iKii

n

iiKii

uNh

uNhu

0,

0,

)(

)()(

pp

source: Mortensonsource: Mortenson

1

1,1

1

1,,

)()()()()(

iki

kiki

iki

kiiki tt

uNut

tt

uNtuuN

ti are knot values that relate u to the control points.

Uniform case: space knots at equal intervals of u.

Repeated knots move curve closer to control points.

N

N

N N

N

N N

N N

Geometric Modeling91.580.201

OpenGL Demoto accompany HW#2