# Epicycloids

If a circle from the radius rolls outside on a circle from the radius , a point on the circumference describes an epicycloid , a special case of a cycloid . ${\ displaystyle a}$ ${\ displaystyle b}$ In this way, mandala-like figures can be drawn that also resemble flowers.

For the mathematical description of an epicycloid one needs - since it is a question of changes in angle - trigonometric expressions.

The equation of an epicycloid is therefore:

${\ displaystyle {\ begin {matrix} x & = & (a + b) \ cdot \ cos (t) -a \ cdot \ cos \ left (\ left (1 + {\ frac {b} {a}} \ right ) t \ right) \\ y & = & (a + b) \ cdot \ sin (t) -a \ cdot \ sin \ left (\ left (1 + {\ frac {b} {a}} \ right) t \ right) \ end {matrix}}}$ It is

• ${\ displaystyle a}$ the radius of the outside moving circle,
• ${\ displaystyle b}$ is the radius of the inner circle,
• ${\ displaystyle t}$ the polar angle of the point where the two circles meet,
• ${\ displaystyle x}$ and the coordinates of the point on the epicycloid.${\ displaystyle y}$ If is an integer, after one revolution we get a closed curve. We set . Then we can write the equation more simply: ${\ displaystyle {\ tfrac {b} {a}}}$ ${\ displaystyle m = 1 + {\ tfrac {b} {a}}}$ ${\ displaystyle {\ begin {matrix} x & = & m \ cdot a \ cdot \ cos (t) -a \ cdot \ cos (m \ cdot t) \\ y & = & m \ cdot a \ cdot \ sin (t) - a \ cdot \ sin (m \ cdot t) \ end {matrix}}}$ The radius R (number b above) in the following diagram is the radius of the large inner circle. The radius of the small circle is r (number a above). On the left it gives a whole number, so the "petals" on the left do not overlap and the curve is closed. On the right, however, the "petals" overlap, i. H. the curve is not closed because = 2.5. is also called the order of the epicycloids: ${\ displaystyle {\ tfrac {R} {r}}}$ ${\ displaystyle {\ tfrac {R} {r}}}$ ${\ displaystyle {\ tfrac {R} {r}}}$ If you are looking for the point opposite the one just drawn and which is an integer, you can give the equation in an alternative form: ${\ displaystyle {\ tfrac {b} {a}}}$ ${\ displaystyle {\ begin {matrix} x & = & m \ cdot a \ cdot \ cos (t) + a \ cdot \ cos (m \ cdot t) \\ y & = & m \ cdot a \ cdot \ sin (t) + a \ cdot \ sin (m \ cdot t) \ end {matrix}}}$ We get the formulas for the length of the epicycloid and the content of the enclosed area

${\ displaystyle {\ begin {matrix} s & = & 8 \ cdot m \ cdot a \\ A & = & m \ cdot (m + 1) \ cdot a ^ {2} \ cdot \ pi \ end {matrix}}}$ Here, too, the tangential vector is normal to the vector . The evolute has the equation ${\ displaystyle TP}$ ${\ displaystyle {\ begin {matrix} x & = & {\ frac {m-1} {m + 1}} \ cdot \ left (m \ cdot a \ cdot \ cos (t) + a \ cdot \ cos (m \ cdot t \ right)) \\ y & = & {\ frac {m-1} {m + 1}} \ cdot \ left (m \ cdot a \ cdot \ sin (t) + a \ cdot \ sin (m \ cdot t \ right)) \ end {matrix}}}$ This is the epicycloid equation in the alternative form, scaled down by the factor . The Evolute is a scaled down, rotated copy of the original curve. ${\ displaystyle {\ tfrac {m-1} {m + 1}}}$ (If the ratio is a rational number, the curve will only close after several revolutions. If it is irrational, it will never close.) ${\ displaystyle {\ frac {b} {a}}}$ An epicycloid is also created by combining two rotations that take place in the same direction of rotation.

## Special epicycloids

For there is a cardioid (heart curve). For the perimeter and area you get: ${\ displaystyle b = a \ (m = 2)}$ ${\ displaystyle s = 16a, \ A = 6a ^ {2} \ pi}$ This curve can also be obtained differently, namely as a conchoidal circle ( Pascal's snail ): You draw a chord from a point on the circumference and extend it by the diameter of the circle. When the tendon rotates, the endpoint of the elongation describes a cardioid.

If the tip of the cardioid is at the origin, the equation in polar or Cartesian coordinates is:

${\ displaystyle {\ begin {matrix} r & = & 2a \ cdot (1+ \ cos (\ varphi)) \\ (x ^ {2} + y ^ {2} -2ax) ^ {2} & = & 4a ^ { 2} \ cdot (x ^ {2} + y ^ {2}) \ end {matrix}}}$ Is , you get a nephroid (kidney curve). She has the dimensions ${\ displaystyle b = 2a \ (m = 3)}$ ${\ displaystyle {\ begin {matrix} s & = & 24a & = & 12b \\ A & = & 12a ^ {2} \ pi & = & 3b ^ {2} \ pi \ end {matrix}}}$ ## literature

• Klemens Burg, Herbert Haf, Friedrich Wille, Andreas Meister: Vector analysis: Higher mathematics for engineers, natural scientists and mathematicians . 2nd edition, Springer, Vieweg-Teubner, Wiesbaden 2012, ISBN 978-3-8348-8346-9 , pp. 56-63.
• Mark J. Wygodski: Higher Mathematics at Hand: Definitions, Theorems, Examples . 2nd edition, Vieweg, Braunschweig 1977, ISBN 3-528-18309-8 , pp. 755-764.
• Matthias Richter: Basic knowledge of mathematics for engineers . Vieweg + Teubner, Wiesbaden 2009, ISBN 978-3-663-05772-7 , pp. 171-172 .