WOLFRAM|DEMONSTRATIONS PROJECT

Trisecting an Angle Using Tschirnhaus's Cubic

​
θ
2.02
length of normal to radius
2
labels
r
3
sec
θ
3
′
r
tan
θ
3
3
sec
θ
3
′
r
r
tan
θ
3
This Demonstration illustrates a property of Tschirnhaus's cubic, which has polar equation
r=
3
sec
θ
3
. Namely, that the angle
SPQ
between the tangent
PS
and the normal
PQ
to the radius vector
OP
at a given point
P
on the curve is one-third of the polar angle
XOP
of the point.
To trisect a given angle
θ
, draw the radius vector (red) from the origin, making that angle with the
x
axis, to meet at a point
P
on the curve. Construct the tangent
PS
(green) and the normal
PQ
to the radius vector (green) at the point. The angle between these two lines is
r'/r=tan(θ/3)
. So the angle
SPQ
is
θ/3
.