[math]r_1[/math]: radius of larger circle[br][math]r_2[/math]: radius of the inner circle[br][br]How can we describe the motion of [math]P[/math] around [math]O[/math]?[br][math]P[/math] starts to the right of [math]O[/math] and moves clockwise around the circle with radius [math]r_2[/math], which can be modeled by the equations:[br] [math]x\left(t\right)=r_2\cos t[/math][br] [math]y\left(t\right)=-r_2\sin t[/math][br][br]How can we describe the motion of [math]O[/math]?[br][math]O[/math] starts on the right and moves counterclockwise around a circle with radius [math]r_1-r_2[/math], which can be modeled by the equations:[br] [math]x\left(t\right)=\left(r_1-r_2\right)\cos t[/math][br] [math]y\left(t\right)=\left(r_1-r_2\right)\sin t[/math][br][br]If you watch carefully, you will see that while [math]O[/math] completes one full rotation for [math]t\in\left[0,2\pi\right][/math], but the number of rotations [math]P[/math] makes around [math]O[/math] is different. We need to adjust the coefficient of [math]t[/math] in our first pair of equations.[br][br]Move the slider to [math]t=0.2\pi[/math]. The arcs highlighted in orange show all of the points on the two circles that have come in contact. The lengths of these arcs are equal. Click the checkbox to show the two central angles for these arcs, [math]\theta_1[/math] and [math]\theta_2[/math]. We can express the arc lengths using the angles and radii.[br] [math]s=r_1\theta_1=r_2\theta_2[/math][br][br]This gives us the relationship: [math]\theta_2=\frac{r_1}{r_2}\theta_1[/math][br][br]At this point, how far has [math]P[/math] rotated around [math]O[/math]? It started from the rightmost point, but [math]\theta_2[/math] is measured from the point of contact. It is too large. Click the other checkbox to show the starting point of [math]P[/math] on the small circle. We can see that the the two horizontal radii are parallel to each other, giving us congruent corresponding angles. The angle between [math]\overline{OP_0'}[/math] and the horizontal radius is also [math]\theta_1[/math]. [br][br]The total rotation of [math]P[/math] around [math]O[/math] can be expressed as:[br] [math]\theta_2-\theta_1=\frac{r_1}{r_2}\theta_1-\theta_1=\left(\frac{r_1-r_2}{r_2}\right)\theta_1[/math][br][br]Since [math]\theta_1=t[/math], we can express this as [math]\left(\frac{r_1-r_2}{r_2}\right)t[/math] and use it to complete our first pair of equations. The motion of [math]P[/math] around [math]O[/math] is given by:[br] [math]x\left(t\right)=r_2\cos\left(\frac{r_1-r_2}{r_2}\right)t[/math][br] [math]y\left(t\right)=-r_2\sin\left(\frac{r_1-r_2}{r_2}\right)t[/math][br]But, since [math]O[/math] itself is moving, we need to add its equations to these. Our final equations for the hypocycloid are:[br] [math]x\left(t\right)=r_2\cos\left(\frac{r_1-r_2}{r_2}\right)t+\left(r_1-r_2\right)\cos t[/math][br] [math]y\left(t\right)=-r_2\sin\left(\frac{r_1-r_2}{r_2}\right)t+\left(r_1-r_2\right)\sin t[/math][br][br]