The Parametric Function Plotter

How to Use

Getting Started

1. Choose Your Mode
   • Click “2D Mode” for planar curves
   • Click “3D Mode” for spatial trajectories

Using 2D Mode

Step 1: Enter Your Functions

In the input fields, enter your parametric equations:

• X(t) = Your function for the x-coordinate (e.g., 16*sin(t)^3)
• Y(t) = Your function for the y-coordinate (e.g., 13*cos(t) - 5*cos(2*t) - 2*cos(3*t) - cos(4*t))

Step 2: Configure Settings

• Time Duration: Set how long the animation should run (in seconds)
• Line Color: Pick your preferred color for the curve

Step 3: Run the Animation

Click the “Run” button to see your function come to life!

What You’ll See:

• Main Plot (XY): The complete parametric curve
• X(t) Plot: How X changes over time
• Y(t) Plot: How Y changes over time

Using 3D Mode

Step 1: Enter Your Functions

Input three parametric equations:

• X(t) = Function for x-coordinate (e.g., cos(t))
• Y(t) = Function for y-coordinate (e.g., sin(t))
• Z(t) = Function for z-coordinate (e.g., t/5)

Step 2: Configure Settings

• Duration: Animation length in seconds
• Line Color: Choose your curve color

Step 3: Run the Visualization

Click “Run” to watch your 3D trajectory unfold!

What You’ll See:

• 3D Trajectory: Interactive 3D view (rotate, zoom, pan)
• X(t) Projection: X component over time
• Y(t) Projection: Y component over time
• Z(t) Projection: Z component over time

Supported Mathematical Functions

Basic Operations

• Addition: +
• Subtraction: -
• Multiplication: *
• Division: /
• Power: ^ or ** (e.g., t^2 or t**2)

Trigonometric Functions

• sin(t) – Sine
• cos(t) – Cosine
• tan(t) – Tangent
• asin(t) – Arcsine
• acos(t) – Arccosine
• atan(t) – Arctangent

Other Functions

• exp(t) – Exponential (e^t)
• log(t) – Natural logarithm
• sqrt(t) – Square root
• abs(t) – Absolute value
• pow(t, n) – Power function

Constants

• pi or PI – Pi (π ≈ 3.14159)
• e or E – Euler’s number (≈ 2.71828)

Example Functions to Try

2D Examples

Heart Curve:

X(t) = 16*sin(t)^3
Y(t) = 13*cos(t) - 5*cos(2*t) - 2*cos(3*t) - cos(4*t)
Duration: 8s

Rose Curve:

X(t) = cos(4*t) * cos(t)
Y(t) = cos(4*t) * sin(t)
Duration: 10s

Lissajous Curve:

X(t) = sin(3*t)
Y(t) = sin(2*t)
Duration: 10s

Spiral:

X(t) = t * cos(t)
Y(t) = t * sin(t)
Duration: 15s

3D Examples

Basic Helix:

X(t) = cos(t)
Y(t) = sin(t)
Z(t) = t/5
Duration: 20s

Trefoil Knot:

X(t) = sin(t) + 2*sin(2*t)
Y(t) = cos(t) - 2*cos(2*t)
Z(t) = -sin(3*t)
Duration: 10s

Toroidal Spiral:

X(t) = (2 + cos(3*t)) * cos(t)
Y(t) = (2 + cos(3*t)) * sin(t)
Z(t) = sin(3*t)
Duration: 10s

Spherical Helix:

X(t) = cos(t) * cos(t/3)
Y(t) = sin(t) * cos(t/3)
Z(t) = sin(t/3)
Duration: 20s

DNA Helix:

X(t) = cos(t)
Y(t) = sin(t)
Z(t) = t/3
Duration: 25s
Color: #00ff00

Tips & Tricks

💡 For Best Results:

• Start with simple functions to understand the behavior
• Experiment with different coefficients to create unique patterns
• Try different colors to highlight different curves
• Adjust duration to see the complete trajectory
• In 3D mode, use your mouse to rotate and explore the curve from different angles

🎨 Creating Beautiful Patterns:

• Combine sine and cosine with different frequencies
• Use prime numbers (2, 3, 5, 7) as coefficients for interesting patterns
• Try nested trigonometric functions like sin(cos(t))
• Experiment with exponential decay: exp(-t/10) * sin(t)

⚠️ Common Mistakes to Avoid:

• Don’t forget the * symbol for multiplication (use 2*t, not 2t)
• Make sure parentheses are balanced
• Avoid division by zero
• Use appropriate duration for your function’s period