Complex Numbers Locus Calculator
Loci and transformations in the complex plane: perpendicular bisectors, circles of Apollonius, circles, ellipses and hyperbolas from |z − a| and |z − b|, arcs from arg((z − a)/(z − b)) = θ, and rotation of a point about another, with exact answers and a sketch.
About the Complex Numbers Locus Calculator
|z − a| is the distance from z to a, so |z − a| = |z − b| is the perpendicular bisector of ab, |z − a| = k|z − b| (k ≠ 1) a circle, |z − a| + |z − b| = 2k an ellipse with foci a and b, and ||z − a| − |z − b|| = 2k a hyperbola. arg((z − a)/(z − b)) = θ is the angle ab subtends at z, so z lies on an arc of a circle through a and b.
Rotating z about a through θ gives z′ = a + (z − a)e^(iθ).
The formula
z′ − a = (z − a)e^(iθ); |z − a| = k|z − b| ⟹ circle (k ≠ 1)
Worked example
|z − (1 + i)| = 2|z − (3 − i)|
- Square: |z − a|² = 4|z − b|²
- Circle centre (11/3, −5/3), radius 4√2/3
Questions people ask
Why is |z − a| = |z − b| a line?
It is the set of points equally far from a and b: the perpendicular bisector of the segment ab.
What does arg((z − a)/(z − b)) mean geometrically?
The angle from the vector z − b to the vector z − a, which is the angle the segment ab subtends at z.