Normals and Tangents of a Trochoid

When the roller is a-rolling, its instantaneous center of rotation is the point P of contact with the supporting circle. That point of the roller is not moving at all; the tracing point is moving with constant distance from that point. Therefore the line from P to the tracing point T is the normal to the trochoid, at T.

In the case that the arm ratio is 1, T is on the rolling circle, and the tangent line passes through the point P′ diametrically opposite to the contact point. Because of this fact, the tangent lines of cycloids have nice properties which are not shared by those of more general trochoids.

I will refer to P′ as the apse of the rolling circle. If R is the radius of the supporting circle and O is its center, then the apse is distant R|1-2w| from O. The circle with that radius and center O I will call the apsidal circle.

Let us remember the Double Generation Theorem. There is an alternative rolling circle, with a contact point Q, through which the normal also passes. Also, when the arm ratio is 1, so that the trochoid is an epicycloid or hypocycloid, the second rolling circle has an apse Q′, which lies on the tangent line.

Here is an interactive display of a trochoid, together with the normal and tangent lines to the tracing point on the rolling circle. Both rolling circles, and their contact points, are shown, color coded as in the display of double generation. The controls of this display are like those of the introductory page on trochoids. When the arm ratio is 1, the display shows the apsidal circle, and the points P′ and Q′.

If N > 1, then there are N lines through the contact point normal to the trochoid, just as there are N points on the roller which can serve as tracers. Moreover, because of double generation, at the alternate contact point there are N′ lines which are also normal to the trochoid.

When the arm ratio is 1, there are similar groups of tangent lines. Specifically, there are N tangent lines through the apse P′, and N′ tangents through the alternate apse Q′.

All of these lines may be seen in the second display. The controls which specify the trochoid are like those of previous displays. One difference is that the knob is attached to the contact point of the rolling circle. Also, there are check boxes which control whether the diagram shows the principal group of normals, the alternate group, or both. When the arm ratio is 1, two more check boxes appear; they control the display of groups of tangents from the apse and alternate apse.

Last modified on $Date: 2015-05-02 12:51:16 -0400 (Sat, 02 May 2015) $

Christopher J. Henrich