GEOMETRIER

For eah desription, provide equation, and parametrization

Circle at origin, radius r

eq: x² + y² = r², Parametrization: (r cos t, r sin t), t ∈ [0, 2π]

Circle at (a, b), radius r

Equation: (x−a)² + (y−b)² = r² — Parametrization: (a + r cos t, b + r sin t), t ∈ [0, 2π]

Ellipse at origin, semi-axes a and b

Equation: x²/a² + y²/b² = 1 — Parametrization: (a cos t, b sin t), t ∈ [0, 2π]

Ellipse at (h, k), semi-axes a and b

Equation: (x−h)²/a² + (y−k)²/b² = 1 — Parametrization: (h + a cos t, k + b sin t), t ∈ [0, 2π]

Line segment from P to Q —

Equation: — — Parametrization: P + t(Q − P), t ∈ [0, 1]

Parabola y = ax² + bx + c

Equation: y = ax² + bx + c — Parametrization: (t, at² + bt + c), t ∈ ℝ

Hyperbola (right branch

Equation: x²/a² − y²/b² = 1 — Parametrization: (a cosh t, b sinh t), t ∈ ℝ

Hyperbola (left branch)

Equation: x²/a² − y²/b² = 1 — Parametrization: (−a cosh t, b sinh t), t ∈ ℝ

Reverse orientation (clockwise instead of CCW)

Flip the sin component sign: (r cos t, −r sin t) for a circle at origin