Important points about the Surface of Revolution
a) if a point on base curve is given by parametric form, that are: (x(u), y(u), z(u)) so surface of revolution regarding to the x-axis will be as:
[x(u), y(u), z(u)] → [x(u), y(u) cos θ, y(u) sin θ] 0 ≤ u ≤ 1; 0 ≤ θ ≤ 2p.
b) Tracing a picture involves movement of points from one position to the other which is the translational transformation is to be utilized. Moving the respective points on base curve from one place to other traces an image, if (x, y, z) is a point on a base curve.
c) If ‾d ⇒ the direction wherein curve is to be shifted and v⇒ scalar quantity representing the amount by that curve is to be moved.
Displacing the curve via amount v ‾d , the curve will be traced on a new position or is swept to a new place.
(x(u), y(u), z(u)) → coordinate points of base curve in parametric form as:
(u → parameter) (x(u), y(u), z(u)) → (x(u), y(u), z(u) ) + v ‾d;
0 ≤ u ≤ 1; 0 ≤ v ≤ 1.
Usually, we can identify sweep constructions by using any path. For rotational sweeps, we can shift along a circular path via any angular distance from 0 to 360^{0}. For noncircular ways, we can identify the curve function explain the path and the distance of travel beside the path. Additionally, we can change the shape or size of the cross section along the sweep way. Or we could change the orientation of the cross section relative to the sweep path like we shift the shape via a region space.