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):
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│ (|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)
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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.
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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.
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Since max(|x|, a) =
╭
│ |x|, if |x|≧a
│ a, if |x|＜a
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and max(|y|, b) =
╭
│ |y|, if |y|≧b
│ b, if |y|＜b,
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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
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