Parametric Equations Solver

Circle

x(t) = r·cos(t)

y(t) = r·sin(t)

• r: radius
• t ∈ [0, 2π]
• Cartesian form: x² + y² = r²

Cycloid

x(t) = r(t - sin(t))

y(t) = r(1 - cos(t))

• r: circle radius
• Generated by point on rolling circle
• Brachistochrone curve

Spiral

x(t) = at·cos(t)

y(t) = at·sin(t)

• a: growth rate
• Archimedean spiral for constant a
• Logarithmic spiral for exponential growth

Lissajous Curves

x(t) = A·sin(at)

y(t) = B·sin(bt + δ)

• A, B: amplitudes
• a, b: frequencies
• δ: phase difference

Arc Length

L = ∫[t₁ to t₂] √[(dx/dt)² + (dy/dt)²] dt

Tangent Vector

T(t) = ⟨dx/dt, dy/dt⟩

Curvature

κ = |x'y'' - y'x''| / (x'² + y'²)^(3/2)

Applications

• Motion Analysis
• Computer Graphics
• Path Planning
• Physics Simulations
• Engineering Design

Vector Analysis

• Position Vector: r(t) = ⟨x(t), y(t)⟩
• Velocity Vector: v(t) = ⟨dx/dt, dy/dt⟩
• Acceleration Vector: a(t) = ⟨d²x/dt², d²y/dt²⟩
• Speed: |v(t)| = √[(dx/dt)² + (dy/dt)²]

Differential Geometry

• Unit Tangent: T(t) = v(t)/|v(t)|
• Normal Vector: N(t) = T'(t)/|T'(t)|
• Binormal Vector: B(t) = T(t) × N(t)
• Torsion: τ = -B'(t)·N(t)

Epitrochoid

x(t) = (R+r)cos(t) - d·cos((R+r)t/r)

y(t) = (R+r)sin(t) - d·sin((R+r)t/r)

• R: fixed circle radius
• r: rolling circle radius
• d: tracing point distance

Cardioid

x(t) = a(2cos(t) - cos(2t))

y(t) = a(2sin(t) - sin(2t))

• Special case of epicycloid
• Heart-shaped curve
• Area = 6πa²

Astroid

x(t) = a·cos³(t)

y(t) = a·sin³(t)

• Four-cusped hypocycloid
• Area = 3πa²/8
• Arc length = 6a

Projectile Motion

x(t) = v₀cos(θ)t

y(t) = v₀sin(θ)t - (1/2)gt²

• v₀: initial velocity
• θ: launch angle
• g: gravitational acceleration

Harmonic Motion

x(t) = A·cos(ωt + φ)

y(t) = B·sin(ωt + φ)

• A, B: amplitudes
• ω: angular frequency
• φ: phase shift
• Applications in vibration analysis

Robotics and Control

• Path planning
• Joint trajectories
• End-effector motion
• Collision avoidance