Definition:Eccentric Angle/Mistake

Source Work

1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.)

2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.):

eccentric angle

Mistake

The angle that a radius of the *auxiliary circle makes with the positive $x$-axis, used in forming the parametric equations of an *ellipse or *hyperbola.

Correction

This definition is too simplistic to be of any use.

A more complete definition of an eccentric angle can be found here:

Let $\KK$ be a central conic.

Let $P$ be a point on $\KK$.

Let $\CC$ be the auxiliary circle of $\KK$.

The eccentric angle of $P$ with respect to $\KK$ is defined as follows:

Ellipse

Let $\KK$ be an ellipse with foci at $F_1$ and $F_2$.

Let $P$ be a point on $\KK$.

Let $\CC$ be the auxiliary circle of $\KK$ with center $O$.

Let $PQ$ be dropped perpendicular to the major axis $F_1 F_2$ of $\KK$ such that $Q$ lies on $F_1 F_2$.

Let $QP$ be produced so as to meet $\CC$ at $P'$.

The angle $QOP'$ is the eccentric angle of $P$ with respect to $\KK$.

In the above diagram, $\alpha$ is the eccentric angle of $P$ with respect to $\KK$.

Hyperbola

Let $\KK$ be a hyperbola with foci at $F_1$ and $F_2$.

Let $P$ be a point on $\KK$.

Let $\CC$ be the auxiliary circle of $\KK$ with center $O$.

Let $PQ$ be dropped perpendicular to the major axis $F_1 F_2$ of $\KK$ such that $Q$ lies on $F_1 F_2$.

Let $QP'$ be drawn tangent to $\CC$ such that $P'$ is the point of tangency with $\CC$.

The angle $QOP'$ is the eccentric angle of $P$ with respect to $\KK$.

In the above diagram, $\alpha$ is the eccentric angle of $P$ with respect to $\KK$.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): eccentric angle
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): eccentric angle