Math Art
Surface of Constant Slope

Surface of Constant Slope

Surface of Constant Slope is a ruled surface, given by a parametrisation, immersed.

Open Surface of Constant Slope in the interactive viewer →

aperiodic def-parametric immersed implemented ruled

Formula

x(u, v)
(1+0.6⁢cos⁡(3⁢u))⁢cos⁡(u)+v⁢(−1.8⁢sin⁡(3⁢u)⁢sin⁡(u)+(1+0.6⁢cos⁡(3⁢u))⁢cos⁡(u))(−1.8⁢sin⁡(3⁢u)⁢cos⁡(u)−(1+0.6⁢cos⁡(3⁢u))⁢sin⁡(u))2+(−1.8⁢sin⁡(3⁢u)⁢sin⁡(u)+(1+0.6⁢cos⁡(3⁢u))⁢cos⁡(u))2\left(1 + 0.6 \cos\left(3 u\right)\right) \cos\left(u\right) + \frac{v \left(-1.8 \sin\left(3 u\right) \sin\left(u\right) + \left(1 + 0.6 \cos\left(3 u\right)\right) \cos\left(u\right)\right)}{\sqrt{\left(-1.8 \sin\left(3 u\right) \cos\left(u\right) - \left(1 + 0.6 \cos\left(3 u\right)\right) \sin\left(u\right)\right)^{2} + \left(-1.8 \sin\left(3 u\right) \sin\left(u\right) + \left(1 + 0.6 \cos\left(3 u\right)\right) \cos\left(u\right)\right)^{2}}}
y(u, v)
(1+0.6⁢cos⁡(3⁢u))⁢sin⁡(u)−v⁢(−1.8⁢sin⁡(3⁢u)⁢cos⁡(u)−(1+0.6⁢cos⁡(3⁢u))⁢sin⁡(u))(−1.8⁢sin⁡(3⁢u)⁢cos⁡(u)−(1+0.6⁢cos⁡(3⁢u))⁢sin⁡(u))2+(−1.8⁢sin⁡(3⁢u)⁢sin⁡(u)+(1+0.6⁢cos⁡(3⁢u))⁢cos⁡(u))2\left(1 + 0.6 \cos\left(3 u\right)\right) \sin\left(u\right) - \frac{v \left(-1.8 \sin\left(3 u\right) \cos\left(u\right) - \left(1 + 0.6 \cos\left(3 u\right)\right) \sin\left(u\right)\right)}{\sqrt{\left(-1.8 \sin\left(3 u\right) \cos\left(u\right) - \left(1 + 0.6 \cos\left(3 u\right)\right) \sin\left(u\right)\right)^{2} + \left(-1.8 \sin\left(3 u\right) \sin\left(u\right) + \left(1 + 0.6 \cos\left(3 u\right)\right) \cos\left(u\right)\right)^{2}}}
z(u, v)
vv

Properties

Family
ruled
Given by
a parametrisation
Curvature
no condition imposed
Periodicity
not periodic
Ends
0
Embedding
immersed
Fidelity
exact
Exactness
elementary

Definition

Monge's sandpile at the operator defaults: rise from the 3-lobed base curve r = 1 + 0.6 cos 3u along its outward normal at 45 degrees (slope 1), ruling length 1

Sources

Chart reproduced numerically against the shipped implementation: fwd max 0.45 of tol, 0.0% of oracle beyond tol (tol floor 2.5 x 0.08839).