Thus, the curve appears as the intersection of two surfaces F(x,y,2)=0∩G(x,y,2)=0 Example: Intersection of the two quadric surfaces z ay and y2= za gives cubic parabola. (These two surfaces intersect not only along the cubic parabola but also along the a-axis. · Explicit curves If the implicit equations can be solved for two of the variables in terms of the third, say for y and z in terms of a, we get y=y( ) z= a(ar) Each of the equations separately represents a cylinder projecting the curve onto one of the coordinate planes. Therefore intersection of the two cylinders represents the curve. Example: Intersection of the two cylinders y=x2, x=x gives a cubic parabola · Parametric curves x=x(t),y=y(t),z=z(t),t1≤t≤t The 3D coordinates (a, y, a) of the point can be expressed on functions of parameter t Here functions a(t), y(t), a(t) have continuous derivatives of the rth order, and the parameter t is restricted to some interval called the parameter space(i.e, tistst2 In this case the curve is said to be of class r, denoted as cr n vector notation where r=(a, y, a), r(t)=(a(t),y(t),a(t)) Example: Cubic parabola Example: Circular helix, see Fig. 2.4 x三aCos (t), y=asin(t),a Using v=tan 2 COS cos t 1+ cos t →COst1- Therefore the following parametrization will give the same circular helix r=a 1+v21+,2btam-1 0≤<∞Thus, the curve appears as the intersection of two surfaces. F(x, y, z) = 0 ∩ G(x, y, z) = 0 Example: Intersection of the two quadric surfaces z = xy and y 2 = zx gives cubic parabola. (These two surfaces intersect not only along the cubic parabola but also along the x-axis.) • Explicit curves If the implicit equations can be solved for two of the variables in terms of the third, say for y and z in terms of x, we get y = y(x), z = z(x) Each of the equations separately represents a cylinder projecting the curve onto one of the coordinate planes. Therefore intersection of the two cylinders represents the curve. Example: Intersection of the two cylinders y = x 2 , z = x 3 gives a cubic parabola. • Parametric curves x = x(t), y = y(t), z = z(t), t1 ≤ t ≤ t2 The 3D coordinates (x, y, z) of the point can be expressed on functions of parameter t. Here functions x(t), y(t), z(t) have continuous derivatives of the rth order, and the parameter t is restricted to some interval called the parameter space (i.e., t1 ≤ t ≤ t2). In this case the curve is said to be of class r, denoted as C r . In vector notation: r = r(t) where r = (x, y, z), r(t) = (x(t), y(t), z(t)) Example: Cubic parabola x = t, y = t 2 , z = t 3 Example: Circular helix, see Fig. 2.4. x = a cos(t), y = a sin(t), z = bt, 0 ≤ t ≤ π Using v = tan t 2 v = tan t 2 = s 1 − cost 1 + cost ⇒ v 2 = 1 − cost 1 + cost ⇒ cost = 1 − v 2 1 + v 2 ⇒ sin t = 2v 1 + v 2 Therefore the following parametrization will give the same circular helix. r = a 1 − v 2 1 + v 2 , 2av 1 + v 2 , 2btan−1 v ! , 0 ≤ v < ∞ 5