Rounded Cuboid with Superquadric Corners

The corners of the rounded cuboids on my previous post are spherical corners.

We can generalize the spherical corners (power = 2) to superquadric corners (power > 0).

Equation 1.

The equation for a rounded cuboid with superquadric corners can be given by:

       max(|x| - a, 0)^p + max(|y| - b, 0)^p + max(|z| - c, 0)^p = d^p,

where a, b, c, d ＞ 0, and p ≧ 1.

p = 1

p = 1.5

p = 2

p = 4

p = 10

Equation 2.

Furthermore, the above equation can be generalized as follows:

       max(|x| - a, 0)^p + max(|y| - b, 0)^q + max(|z| - c, 0)^r = d,

where a, b, c ≧ 0, d ＞ 0, and p, q, r ＞ 0.

Related posts:

• Equation of a Rounded Rectangle

Equation of a Rounded Rectangle

Equations:

The equation of a rounded rectangle centered at (0,0) with width 2(a+c), height 2(b+c)
and corner radius c can be expressed as

       max(|x| - a, 0)² + max(|y| - b, 0)² = c² ...... (I)

or

       (|x| - a + │|x| - a│)² + (|y| - b + │|y| - b│)² = (2c)² ...... (II)

, where a, b, and c are positive numbers.

Proof:

The rounded rectangle can be broken down into four parts (see the image below):
       ╭
       │ (|x| - a)² + (|y| - b)² = c²,  if |x|≧a and |y|≧b ...... (1)
       │ |x| = a + c                  ,  if |x|≧a and |y|＜b ...... (2)
       │ |y| = b + c                  ,  if |x|＜a and |y|≧b ...... (3)
       │ no graph                    ,  if |x|＜a and |y|＜b ...... (4)
       ╰

Since |x|≧a in part (2) and |y|≧b in part (3),
the above expression is equivalent to
       ╭
       │ (|x| - a)² + (|y| - b)² = c²,  if |x|≧a and |y|≧b;
       │ (|x| - a)² = c²               ,  if |x|≧a and |y|＜b;
       │ (|y| - b)² = c²               ,  if |x|＜a and |y|≧b;
       │ 0 = c²                        ,  if |x|＜a and |y|＜b.
       ╰

Now make the equations of the four parts look similar:
       ╭
       │ (|x| - a)² + (|y| - b)² = c²,  if |x|≧a and |y|≧b;
       │ (|x| - a)² +   (b - b)² = c²,  if |x|≧a and |y|＜b;
       │   (a - a)² + (|y| - b)² = c²,  if |x|＜a and |y|≧b;
       │   (a - a)² +   (b - b)² = c²,  if |x|＜a and |y|＜b.
       ╰

Since max(|x|, a) =
               ╭
               │ |x|,  if |x|≧a
               │  a,  if |x|＜a
               ╰
and max(|y|, b) =
               ╭
               │ |y|,  if |y|≧b
               │  b,  if |y|＜b,
               ╰
the four equations can be combined into a single one as follows:

       (max(|x|, a) - a)² + (max(|y|, b) - b)² = c²,

or equivalently,

       max(|x| - a, 0)² + max(|y| - b, 0)² = c² ...... (I)


Using the formula max(z, 0) = (z + |z|) / 2,
the above equation can be expanded to

       (|x| - a + │|x| - a│)² + (|y| - b + │|y| - b│)² = (2c)² ...... (II)

Note:

• When a or b is zero (but not both), the equation (I) describes a stadium (curve).
• When c is zero (a and b are positive), the equation (I) describes a rectangular region rather than a rectangle.
• The equation  max(|x| - a, |y| - b) = 0  describes a rectangle.
• The equation  max(|x| - a, 0)² + max(|y| - b, 0)² + max(|z| - c, 0)² = d²  describes a rounded cuboid.

max(|x|, 0)² + max(|y| - 1, 0)² = 2²

(a = 0)

max(|x| - 3, 0)² + max(|y|, 0)² = (1.5)²

(b = 0)

max(|x| - 3, 0)² + max(|y| - 2, 0)² = 0

(c = 0)

max(|x| - 4, 0)² + max(|y| - 3, 0)² + max(|z| - 1, 0)² = 1

Related posts:

• Rounded Cross
• Rounded Cuboid with Superquadric Corners

Funny Boxing Champion (2)

( Mathematical software used: gnuplot )

Funny Boxing Champion (1)

Equations (方程式):


   (1) body (身体):

x² + y² - 72[exp(1/(sin(y+x) - 1)) ± exp(1/(sin(y-x) - 1))]² = 5/2

   (2) eye sockets (眼眶):

(abs(x) - π/4)² + (y - 3π/2)² = (π/8)²

   (3) eyes (眼睛):

arcsin²(k(abs(x) - π/4)) = arcsin²(k(y - 3π/2))

   (4) mouth (嘴巴):

arcsin²(kx) = arcsin²(k(y - 19π/16))

   (5) navel (肚臍):

arcsin²(kx) = arcsin²(k(y + π/3))

(8 ≦ k ≦ 10)

( Mathematical software used: gnuplot )