## Monday, October 17, 2016

### 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