Module revolve 

Revolution of a profile about an axis.

The profile is flattened to rings in its own plane, then each ring point is rotated about the axis in steps chosen by the chord budget. A full turn closes on itself; a partial turn is capped by the profile at each end.

Pappus gives the oracle: a full revolution has volume 2*pi*R*A, where R is the centroid’s distance from the axis and A the profile area.

How far the walls sit from the surface (#231)

revolve bounds the distance from every point of the surface swept by the GIVEN rings to its triangles by tolerance.linear(). Rings that chord a curved profile add their own deviation on top: a caller that wants the exact surface of revolution within a budget c flattens the profile to c_p and revolves to c - c_p. The reference compiler takes half each, which minimises the triangle count for a fixed sum: the steps round the profile and round the axis each grow as the inverse square root of their share, so their product is smallest at an even split. The two shares add because a rotation is an isometry: a point of the exact surface is the rotation of a profile point within c_p of the ring polygon, so it is within c_p of the same rotation of the polygon.

Each ring segment q_a -> q_b sweeps, between the stations at angles t and t + h, the surface R(s) q(x) with q(x) = (1-x) q_a + x q_b. For fixed x the point R(s) q(x) runs along an arc of radius rho(x), its distance from the axis, and the chord between the arc’s ends is the row at x of the bilinear patch over the quad R(t) q_a, R(t) q_b, R(t+h) q_a, R(t+h) q_b (a rotation is linear in the point). An arc of angle h stays within its sagitta rho (1 - cos(h/2)) of its chord, and rho(x) is the norm of an affine function, so convex in x and largest at a ring vertex: every point of the swept surface is within rho_max (1 - cos(h/2)) of the patch, with rho_max the largest vertex distance from the axis over every ring – the outer equator, for a torus, never the tube centre. The patch is in turn within loft::quad_deviation of the two triangles the loft cuts it into. When the axis lies in the profile’s plane every quad is a planar trapezoid (its two chords are images of parallel radial vectors under the same R(t+h) - R(t)), convex while the segment stays on one side of the axis, and that term is zero: the bound is the sagitta alone, whatever the sign of the Gaussian curvature (the inner side of a torus is saddle-shaped, but its trapezoids are still flat). An axis off the profile’s plane sweeps hyperboloidal quads; their measured twist is added, and the step shrinks until the sum fits.

Every span of a revolution is the first span rotated, so the twist is measured once.

Functions

revolve

Revolve a profile about an axis into a closed solid.