Parametric Curves & Surfaces
Parametric Curves & Surfaces Parametric / Implicit Parametric Curves (Splines) Parametric Surfaces Subdivision Marc Neveu ± LE2I ± Université de Bourgogne 1 2 ways to define a circle Parametric Implicit F>0 F=0 u F<0 xu=r.cosu F x ,y= x2 y2−r 2 {yu=r.sinu 2 - Representations of a curve (2D) Explicit : y = y(x) y=mx+b y=x2 must be a function (x->y) : important limitation Parametric : (x,y)=x(u),y(u)) (x,y) = (cos u, sin u) + easy to specify - hidden additional variable u : the parameter Implicit : f(x,y) = 0 x2+y2-r2=0 + y can be a multivalued function of x - difficult to specify, modify, control 3 - Representations of curves and surfaces (3D) x2+y2+z2-1=0 · Implicit Representation x-y=0 · Curves : f(x,y,z) = 0 and g(x,y,z) =0 · Surfaces : f(x,y,z) =0 · Parametric Representation · Curves : x = f(t), y = g(t), z = h(t) · Surfaces : x = f(u,v), y = g(u,v), z = h(u,v) · Cubic Curve : 3 2 · x= axt + bxt + cxt + dx 3 2 · y = ayt + byt + cyt + dy 3 2 · z = azt + bzt + czt + dz 4 xt =x R.cos2.t 0 Helicoid : yt =y R.sin2 .t 0 u∈ℝ ,v∈ℝ = {z t z0 p.t x=v.cosu y=v.sinu Helix passing by (x ,y ,z ), radius R, 0 0 0 {z=u pitch p 5 Representation of Surfaces Parametric Surface ± (x(u,v),y(u,v),z(u,v)) Ex : plane, sphere, cylinder, torus, bicubic surface, sweeping surface parametric functions allow moving onto the surface modifying u and v (eg in nested loops) Easy for polygonal meshes, etc Terrific for intersections (ray/surface), testing points inside/border, etc Implicit Surface F(x,y,z)=0 Ex : plane, sphere, cylinder, torus, blobs Terrific for
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