Introduction to Quaternions

36. Equations of the circle.

Let AD be the diameter of the circle, center C, radius = a, P any point.

If vector CD = α, CP = ρ, we have

  ρ² = -a²                      (1). 

If however AP = ρ, CP = ρ - α, we have

  (ρ - α)² = -a²                (2). 

If O be any point,

  OP = ρ, OC = γ, CP = ρ - γ,

  (ρ - γ)² = -a²                (3).

These are the three forms of the vector equation.

Form (2) may be written

  ρ² - 2Sαρ = 0.

If OC = c, form (3) may be written

  ρ² - 2Sγρ = c² - a².

Examples.

37. EX. 1. The angle in a semicircle is a right angle.

Taking the second form

  ρ² - 2Sαρ = 0,

  Sρ(ρ - 2α) = 0;