Parametric Point on the Ellipse

θ is the eccentric angle — not the actual angle subtended at the centre. Every point on the ellipse corresponds to a unique θ. The parametric form simplifies tangent, normal, and focal chord calculations.

Set $x = a\cos\theta$, $y = b\sin\theta$. Substituting into $x^2/a^2 + y^2/b^2$:

$$\frac{a^2\cos^2\theta}{a^2} + \frac{b^2\sin^2\theta}{b^2} = \cos^2\theta + \sin^2\theta = 1 \checkmark$$

Every value of $\theta \in [0°, 360°)$ gives a distinct point on the ellipse.

Eccentric angle vs actual angle: $\theta$ is NOT the angle $\angle POx$ where $P = (a\cos\theta, b\sin\theta)$. The actual angle is $\phi = \arctan(y/x) = \arctan\!\left(\frac{b}{a}\tan\theta\right)$. They coincide only at $\theta = 0°, 90°, 180°, 270°$.

Chord joining two parametric points: The chord joining $(a\cos\alpha, b\sin\alpha)$ and $(a\cos\beta, b\sin\beta)$ has equation:

$$\frac{x}{a}\cos\!\frac{\alpha+\beta}{2} + \frac{y}{b}\sin\!\frac{\alpha+\beta}{2} = \cos\!\frac{\alpha-\beta}{2}$$

Focal distances in terms of $\theta$:

$$r_1 = a - e \cdot a\cos\theta = a(1 - e\cos\theta)$$ $$r_2 = a + e \cdot a\cos\theta = a(1 + e\cos\theta)$$