Hypotrochoïde

In Geometry, the hypotrochoïdes are plane Courbe S described by a point related to a mobile circle (C) travelling without slipping on and internally at a basic circle (C0), the travelling circle being smaller than fixes it. These curves were studied by Dürer in 1525, Romer in 1674 and Bernoulli in 1725:

The word is composed of the Greek roots hupo (below) and trokhos (the wheel). When the circle rolls outside, one deals with épitrochoïde.

Parameter setting

One thus poses $q=a/b$ (q>1) and d=k*b, with has the ray of the fixed circle and B is that of the travelling circle (mobile) and D the distance from the center point of the mobile circle. A parameter setting (given in affix) of the hypotrochoïde is then:

$z=\; (a-b)\; exp\; (it)\; +dexp\; (-\; I\; (q-1)\; T)$

that is to say

$qz=a\; ((q-1)\; exp\; (it)\; +kexp\; (-\; it\; (q-1))$

$q\; (x+iy)\; =\; has\; (q-1)\; cos\; (T)\; +\; ia\; (q-1)\; sin\; (T)\; +\; akcos\; ((q-1)\; T)\; -\; iaksin\; ((q-1)\; T)$

By identification of the parts real and imaginary one obtains:

$qx\; =\; has\; (q-1)\; cos\; (T)\; +kacos\; ((q-1)\; T));$

$qy\; =\; has\; (q-1)\; sin\; (T)\; -\; kasin\; ((q-1)\; T));$

with q=a/b and k=d/b

If one poses has = R and b=r and $t=\; \backslash \; theta$, one obtains the formles below:

$x\; =\; (R\; -\; R)\; \backslash \; cos\; \backslash \; theta\; +\; D\; \backslash \; cos\; \backslash \; left\; (\{R\; -\; R\; \backslash \; over\; R\}\; \backslash \; theta\; \backslash \; right)$

$y\; =\; (R\; -\; R)\; \backslash \; sin\; \backslash \; theta\; -\; D\; \backslash \; sin\; \backslash \; left\; (\{R\; -\; R\; \backslash \; over\; R\}\; \backslash \; theta\; \backslash \; right)$

the varying parameter $\backslash \; theta$ of 0 with 2*Pi.

The Hypocycloïde S represent the particular case D = R (the fixed point is on the circle) and the ellipse S the case R = 2r .

Definition of surface, textures and traced for Mathematica 5.2

Parameter setting of surface

One introduces a second parameter in the following way:

$x\; =\; sin\; (v)\; (has\; ((1-1/q)\; cos\; (U)\; +\; (k/q)\; cos\; (q-1)\; U));$

$y=sin\; (v)\; (has\; ((1-1/q)\; sin\; (U)\; -\; (k/q)\; sin\; ((q-1)\; U));$

$z=ccos\; (v);$

hupoTrokhosDisq c_, k_, q_ v_ = {Sin* (a* ((1 - 1/q) *Cos + (k/q) *Cos - 1) *u)), Sin* (a* ((1 - 1/q) *Sin - (k/q) *Sin - 1) *u)), c*Cos};

Parameter setting of surface (with texture)

Parameter setting of texture

hupoTrokhosDisque c_, k_, q_ v_ = Modulate {red, purple}, Suspends hupoTrokhosDisq C, K, Q v, {RGBColor = ABS], purple = ABS], 1 - red*purple], EdgeForm}]];

Parameter setting of the layout

hupoTrokhosGraph c_, k_, q_ v_: = ParametricPlot3D Evaluate C, K, Q v], {U, 0,2Pi, Pi/80}, {v, 0, pi, Pi/100}, ImageSize - > {di, di}, Boxed - > False, Axes - > False, ViewPoint - > {0, 0,8}, Lighting - > False, Epilog - > {Text " , {0.85, 0.95}, Text {0.95, 0.95}], Text " , {0.85, 0.95}, Text {0.85, 0.95}}];

GraphicsArray K, Q v, {Q, 2,9}, {K, 0.25,2,0.5}]];

See too

• Épitrochoïde
• Hypocycloid

Random links:

Paul Jullemier | Gold medal of the Canadian Council of the engineers | Mourtala Diakité | Colli Etruschi Viterbesi rosato | Marcus Horatius Barbatus | Helmuts_Balderis

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