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Parallel Curves

Named curves

circle parallel curve | ellipse parallel curve | parabola parallel curve

Example plots

Example plots Circle parallel curve

Example plots Ellipse parallel curve

Example plots Parabola parallel curve

Equations

circle parallel curve | x(t) = (a + k) cos(t) y(t) = (a + k) sin(t) ellipse parallel curve | x(t) = cos(t) ((b k)/sqrt(a^2 sin^2(t) + b^2 cos^2(t)) + a) y(t) = sin(t) ((a k)/sqrt(a^2 sin^2(t) + b^2 cos^2(t)) + b) parabola parallel curve | x(t) = 2 a t + (k t)/sqrt(t^2 + 1) y(t) = a t^2 - k/sqrt(t^2 + 1)

circle parallel curve | x^2 + y^2 = (a + k)^2 ellipse parallel curve | b^4 x^8 + 2 b^6 x^6 + 2 y^2 b^4 x^6 - 4 a^2 b^4 x^6 + 2 y^2 a^2 b^2 x^6 - 4 b^4 k^2 x^6 + 2 a^2 b^2 k^2 x^6 + b^8 x^4 - 2 y^2 b^6 x^4 - 6 a^2 b^6 x^4 + y^4 a^4 x^4 + y^4 b^4 x^4 + 6 a^4 b^4 x^4 + 2 y^2 a^2 b^4 x^4 + a^4 k^4 x^4 + 6 b^4 k^4 x^4 - 6 a^2 b^2 k^4 x^4 - 6 y^2 a^4 b^2 x^4 + 4 y^4 a^2 b^2 x^4 - 6 b^6 k^2 x^4 - 2 y^2 a^4 k^2 x^4 - 6 y^2 b^4 k^2 x^4 + 10 a^2 b^4 k^2 x^4 - 6 a^4 b^2 k^2 x^4 + 2 y^2 a^2 b^2 k^2 x^4 - 2 a^2 b^8 x^2 - 2 y^4 a^6 x^2 + 6 a^4 b^6 x^2 + 6 y^2 a^2 b^6 x^2 - 2 a^4 k^6 x^2 - 4 b^4 k^6 x^2 + 6 a^2 b^2 k^6 x^2 + 2 y^6 a^4 x^2 - 4 a^6 b^4 x^2 - 10 y^2 a^4 b^4 x^2 - 6 y^4 a^2 b^4 x^2 - 2 a^6 k^4 x^2 + 6 b^6 k^4 x^2 + 6 y^2 a^4 k^4 x^2 + 6 y^2 b^4 k^4 x^2 - 8 a^2 b^4 k^4 x^2 + 4 a^4 b^2 k^4 x^2 - 10 y^2 a^2 b^2 k^4 x^2 + 6 y^2 a^6 b^2 x^2 + 2 y^4 a^4 b^2 x^2 + 2 y^6 a^2 b^2 x^2 - 2 b^8 k^2 x^2 + 4 y^2 a^6 k^2 x^2 + 4 y^2 b^6 k^2 x^2 + 4 a^2 b^6 k^2 x^2 - 6 y^4 a^4 k^2 x^2 - 2 y^4 b^4 k^2 x^2 - 8 a^4 b^4 k^2 x^2 - 6 y^2 a^2 b^4 k^2 x^2 + 6 a^6 b^2 k^2 x^2 - 6 y^2 a^4 b^2 k^2 x^2 + 2 y^4 a^2 b^2 k^2 x^2 + y^4 a^8 + a^4 b^8 + a^4 k^8 + b^4 k^8 - 2 a^2 b^2 k^8 + 2 y^6 a^6 - 2 a^6 b^6 - 4 y^2 a^4 b^6 - 2 a^6 k^6 - 2 b^6 k^6 - 4 y^2 a^4 k^6 - 2 y^2 b^4 k^6 + 2 a^2 b^4 k^6 + 2 a^4 b^2 k^6 + 6 y^2 a^2 b^2 k^6 + y^8 a^4 + a^8 b^4 + 6 y^2 a^6 b^4 + 6 y^4 a^4 b^4 + a^8 k^4 + b^8 k^4 + 6 y^2 a^6 k^4 - 2 y^2 b^6 k^4 + 2 a^2 b^6 k^4 + 6 y^4 a^4 k^4 + y^4 b^4 k^4 - 6 a^4 b^4 k^4 + 4 y^2 a^2 b^4 k^4 + 2 a^6 b^2 k^4 - 8 y^2 a^4 b^2 k^4 - 6 y^4 a^2 b^2 k^4 - 2 y^2 a^8 b^2 - 6 y^4 a^6 b^2 - 4 y^6 a^4 b^2 - 2 y^2 a^8 k^2 - 2 a^2 b^8 k^2 - 6 y^4 a^6 k^2 + 2 a^4 b^6 k^2 + 6 y^2 a^2 b^6 k^2 - 4 y^6 a^4 k^2 + 2 a^6 b^4 k^2 - 8 y^2 a^4 b^4 k^2 - 6 y^4 a^2 b^4 k^2 - 2 a^8 b^2 k^2 + 4 y^2 a^6 b^2 k^2 + 10 y^4 a^4 b^2 k^2 + 2 y^6 a^2 b^2 k^2 = 0 parabola parallel curve | 16 a^4 k^2 - 16 a^4 y^2 - 32 a^3 k^2 y + 8 a^3 x^2 y + 32 a^3 y^3 + 8 a^2 k^4 + 20 a^2 k^2 x^2 + 8 a^2 k^2 y^2 - a^2 x^4 - 32 a^2 x^2 y^2 - 16 a^2 y^4 - 8 a k^4 y - 2 a k^2 x^2 y + 8 a k^2 y^3 + 10 a x^4 y + 8 a x^2 y^3 + k^6 - 3 k^4 x^2 - k^4 y^2 + 3 k^2 x^4 + 2 k^2 x^2 y^2 - x^6 - x^4 y^2 = 0

circle parallel curve | r(θ) = a + k

Common properties

algebraic | parallel curve | parametric

Basic properties

circle parallel curve | A = π (a + k)^2 ellipse parallel curve | A = 2 a k E(1 - b^2/a^2) + 2 b k E(1 - a^2/b^2) + π (a b + k^2)

circle parallel curve | s = 2 π (a + k)

circle parallel curve | d = 2 ellipse parallel curve | d = 8 parabola parallel curve | d = 6

Conic properties

| circle parallel curve eccentricity | e = 0

Derived curves

| circle parallel curve | parabola parallel curve evolute | point at origin | point at origin

Distance properties

| circle parallel curve mean line segment length | s^_ = (4 (a + k))/π

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