- We can think of the derivative[dx(t)dtdP(t)dy(t)dtdtdz(t)dtAs the velocity with which the curve is traced outand points in direction tangent to the curve
– We can think of the derivative As the velocity with which the curve is traced out and points in direction tangent to the curve. ( ) ( ) ( ) ( ) dx t dt dP t dy t dt dt dz t dt
Parametric polynomial curves: Parametric curves are not unique. A given curve orsurface can be represented in many ways, but weshall find that parametric forms in which thefunctions are polynomials in t.: A polynomial parametric curve of degree 3 is offormP(t) = at3 + bt? +ct +dIt was known as Ferguson curve, and was used forairplane design in earlier in USA
Parametric polynomial curves • Parametric curves are not unique. A given curve or surface can be represented in many ways, but we shall find that parametric forms in which the functions are polynomials in t. • A polynomial parametric curve of degree 3 is of form • It was known as Ferguson curve, and was used for airplane design in earlier in USA. 3 2 P t at bt ct d ( )
: But Ferguson curve is not straightforward.even the coefficients a, b, c, d are given, it'sstill hard to imagine the shape of the curve
• But Ferguson curve is not straightforward, even the coefficients a, b, c, d are given, it’s still hard to imagine the shape of the curve
Bezier.Cuvres:Conceptandproperties· Pierre Bezier proposed a method to representa curve in terms of connected vectors.V(t)=Zf..(t)A,i=01, i=0,fin(t)=di-1(1-t)n-1(-t)(i-1)! dti-i
Bézier Cuvres: Concept and properties • Pierre Bézier proposed a method to represent a curve in terms of connected vectors. , 0 1 1 , 1 ( ) ( ) 1, 0, ( ) ( ) (1 ) 1 ( 1)! n i n i i i i n i n i V t f t A i f t t d t i dt t
: Curve can be designed interactively?- Curve are generated according to the controlnet (those connected vectors)- We may change vectors to modify the curve-Seedemo:curvesystem.exe
• Curve can be designed interactively? – Curve are generated according to the control net (those connected vectors) – We may change vectors to modify the curve – See demo: curvesystem.exe