Parametric Equations Solver

Circle

x(t) = r·cos(t)

y(t) = r·sin(t)

• r: radius - This value determines the size of the circle.
• t ∈ [0, 2π] - The parameter 't' typically ranges from 0 to 2π (or 0 to 360 degrees) to complete one full revolution of the circle.
• Cartesian form: x² + y² = r² - This is the familiar equation of a circle in standard (Cartesian) coordinates, showing the direct relationship between the parametric and Cartesian forms.

Cycloid

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

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

• r: circle radius - This is the radius of the rolling circle that generates the cycloid.
• Generated by point on rolling circle - A cycloid is the curve traced by a point on the circumference of a circle as it rolls along a straight line without slipping.
• Brachistochrone curve - The cycloid is famous for being the curve of fastest descent (brachistochrone) and the curve of constant descent (tautochrone), meaning an object sliding down it will reach the bottom in the same amount of time regardless of its starting point.

Spiral

x(t) = at·cos(t)

y(t) = at·sin(t)

• a: growth rate - This constant 'a' controls how quickly the spiral expands or contracts as 't' increases.
• Archimedean spiral for constant a - When 'a' is a constant, this equation describes an Archimedean spiral, where the distance between successive turns is constant.
• Logarithmic spiral for exponential growth - Other types of spirals, like the logarithmic spiral, involve exponential growth and are found frequently in nature (e.g., nautilus shells, hurricane patterns).

Lissajous Curves

x(t) = A·sin(at)

y(t) = B·sin(bt + δ)

• A, B: amplitudes - These values determine the maximum extent of the curve along the x and y axes, respectively.
• a, b: frequencies - These constants represent the frequencies of the sinusoidal oscillations along the x and y axes. The ratio a/b determines the complexity and pattern of the curve.
• δ: phase difference - This value shifts the starting point of the y-oscillation relative to the x-oscillation, significantly altering the curve's shape.

Arc Length

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

The arc length formula calculates the total distance along the curve between two specific parameter values (t₁ and t₂). It's like measuring the length of a string laid along the curve.

Tangent Vector

T(t) = ⟨dx/dt, dy/dt⟩

The tangent vector gives the direction and magnitude of the curve's movement at any given point 't'. It represents the instantaneous velocity if 't' is time.

Curvature

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

Curvature (κ) measures how sharply a curve bends at a particular point. A high curvature means a tight bend, while a low curvature indicates a straighter path.

Applications

• Motion Analysis: Used to track the path and speed of objects, like projectiles or planets.
• Computer Graphics: Essential for drawing smooth curves and complex shapes in animation and design software.
• Path Planning: Helps robots and autonomous vehicles navigate by defining precise routes.
• Physics Simulations: Models the movement of particles and forces in various physical systems.
• Engineering Design: Applied in designing gears, cams, and other mechanical components that require precise curved movements.

Vector Analysis

• Position Vector: r(t) = ⟨x(t), y(t)⟩ - This vector points from the origin to any point on the curve at a given parameter 't'.
• Velocity Vector: v(t) = ⟨dx/dt, dy/dt⟩ - The first derivative of the position vector, it indicates the instantaneous speed and direction of motion along the curve.
• Acceleration Vector: a(t) = ⟨d²x/dt², d²y/dt²⟩ - The second derivative of the position vector, it describes how the velocity vector is changing (i.e., the rate of change of speed or direction).
• Speed: |v(t)| = √[(dx/dt)² + (dy/dt)²] - The magnitude of the velocity vector, representing the instantaneous speed of the object moving along the curve.

Differential Geometry

• Unit Tangent: T(t) = v(t)/|v(t)| - A vector of length one that points in the direction of the curve's motion at any point.
• Normal Vector: N(t) = T'(t)/|T'(t)| - A vector perpendicular to the unit tangent, pointing towards the concave side of the curve, indicating the direction of bending.
• Binormal Vector: B(t) = T(t) × N(t) - For 3D curves, this vector is perpendicular to both the tangent and normal vectors, completing an orthonormal frame (TNB frame) that moves along the curve.
• Torsion: τ = -B'(t)·N(t) - A measure of how much a 3D curve twists out of its osculating plane (the plane containing the tangent and normal vectors).

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 - The radius of the larger, stationary circle.
• r: rolling circle radius - The radius of the smaller circle that rolls around the outside of the fixed circle.
• d: tracing point distance - The distance of the tracing point from the center of the rolling circle. Epitrochoids create complex, often flower-like patterns, and include shapes like the rose curve and spirograph patterns.

Cardioid

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

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

• Special case of epicycloid - A cardioid is a specific type of epicycloid where the rolling circle has the same radius as the fixed circle.
• Heart-shaped curve - Its name comes from its distinctive heart-like shape.
• Area = 6πa² - The area enclosed by a cardioid can be calculated using this formula, where 'a' is a scaling factor.

Astroid

x(t) = a·cos³(t)

y(t) = a·sin³(t)

• Four-cusped hypocycloid - An astroid is a hypocycloid (a curve traced by a point on a circle rolling inside another circle) with four cusps (sharp points).
• Area = 3πa²/8 - The area enclosed by the astroid.
• Arc length = 6a - The total length of the astroid's curve.

Projectile Motion

x(t) = v₀cos(θ)t

y(t) = v₀sin(θ)t - (1/2)gt²

• v₀: initial velocity - The starting speed of the object.
• θ: launch angle - The angle at which the object is launched relative to the horizontal.
• g: gravitational acceleration - The constant acceleration due to gravity (approx. 9.81 m/s² on Earth). These equations describe the parabolic path of an object under gravity, ignoring air resistance.

Harmonic Motion

x(t) = A·cos(ωt + φ)

y(t) = B·sin(ωt + φ)

• A, B: amplitudes - The maximum displacement from the equilibrium position along the x and y axes.
• ω: angular frequency - How fast the oscillation occurs.
• φ: phase shift - Determines the starting position of the oscillation.
• Applications in vibration analysis - These equations are used to model oscillating systems, such as pendulums, springs, and sound waves.

Robotics and Control

• Path planning - Parametric equations define smooth and efficient paths for robots to follow.
• Joint trajectories - They describe the precise movement of each joint in a robotic arm.
• End-effector motion - Used to control the exact position and orientation of the robot's gripper or tool.
• Collision avoidance - By accurately modeling robot movement, parametric equations help prevent collisions with obstacles or other robots.