1. Curve & Surface

2. Three basic forms of curves (1/4)
Explicit form y = f(x)
impossible to get multiple values for a single x
break curves like circles and ellipses into segments
not invariant with rotation
rotation might require further segment breaking
problem with curves with vertical tangents
infinite slope is difficult to represent
Implicit form f(x, y, z) = 0
equation may have more solutions than we want
circle: x² + y² = 1, half circle: ?
problem to join curve segments together
difficult to determine if their tangent directions agree at their joint point
So, we use parametric form for curves and surfaces

3. Three basic forms of curves (2/4)
Parametric form : x = x(t), y = y(t), z = z(t)
overcomes problems with explicit and implicit forms
no geometric slopes (which may be infinite)
parametric tangent vectors instead (never infinite)
a curve is approximated by a piecewise polynomial curve

4. Three basic forms of curves (3/4)
Parametric form : based on the curve length
For parameter t, we obtain equation
we denote x=x(t), y=y(t)
position vector p(t)=[x(t), y(t)]
derivative
slope of curve =
axis independent because point on a curve is specified by a single value of parameter t.
end point and length are fixed by parameter range
usually normalized to

5. Three basic forms of curves (4/4)
Ex) parametric rep. Of straight line from position vector to
- each components of P(t) has a parametric representation
when
slope

6. Parametric Curve (1/6) Three Major types of Curves Hermite Bezier
Two end points & two end point tangent vectors
Bezier
Defined by two end points and two other points that control the end point tangent vectors
Spline
several kinds, each defined by four points
uniform B-splines, non-uniform B-splines, ß-splines

7. Parametric Curve (2/6) Hermite Curve Bezier Curve
Two end points & two end point tangent vectors
Bezier Curve
two end points and two other points that control the end point tangent vectors

8. Parametric Curve (3/6) Bezier curve properties
The first and last control points are the end points of the curve segment
Convexhull property
The curve is contained in shaded area formed from the control points
The control points do not exert ‘local’ control
Moving any control point affect all of the curve to a greater or lesser extent

9. Parametric Curve (4/6) B-spline curve property four control points
convex hull property
The curve is transformed by applying affine transformation
A B-spline curve exhibit local control

10. Parametric Curve (5/6) Geometric continuity Parametric continuity
G0 geometric continuity
two curve segment join together
G1 geometric continuity
the directions of the two segment’s tangent vectors are equal at a join point
not necessarily the magnitudes
Parametric continuity
C1parametric continuity
tangent vectors of two curve segments are equal at the join point
direction and magnitude
Cn parametric continuity
the direction and magnitude of dn/dtn[Q(t)] through the nth derivative are equal at the join point

11. Parametric Curve (6/6) Subdivision Purpose :
To render a curve, rather than evaluate points along a curve, it is more cheaper to subdivide recursively until the convex hulls are sufficiently good approximation to the curve.
The termination criterion.
linearity test applied to the convex hull. i.e., how far the interior control points deviate from the line connecting the two outer control points.

12. Biparametric patch (1/3)
Each patch is defined by blending control points

13. Biparametric patch (2/3)

14. Biparametric patch (3/3)
Joining Two patch
C0 continuity requires aligning boundary curves
C1 continuity requires aligning boundary curves and derivatives

15. Parametric Surfaces Advantages: Disadvantages:
Easy to enumerate points on surface
Possible to describe complex shapes
Disadvantages:
Control mesh must be quadrilaterals
Continuity constraints difficult to maintain:
C0 easy, C1 possible, C2 hard at extraordinary vertices
Hard to find intersections