Curve & Surface

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

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

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

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

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

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

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

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

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

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.

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

Biparametric patch (2/3)

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

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