Develop a mathematical formula that can generate polygons with variable curvature near vertices, transitioning between sharp polygon edges and smooth circular shapes | Step-by-Step Solution

Let's Create Smooth Polygons! 🔄

What We're Solving:

We want to create a mathematical formula that can generate shapes that smoothly transition from sharp-cornered polygons (like triangles, squares, pentagons) to perfect circles, with a parameter that controls how "rounded" the corners become.

The Approach:

We'll use a "smoothness dial" - when it's at 0, we get sharp polygon corners, and when we turn it up, the corners gradually become more rounded until we eventually get a circle! We'll use parametric equations and a technique called "superellipse" or "Lamé curves."

Step-by-Step Solution:

Step 1: Start with the basic idea

• A regular N-sided polygon has vertices equally spaced around a circle
• A circle is just the limiting case where we have infinite "sides"
• We need a way to "blend" between these two shapes

Step 2: Use the superellipse concept The key insight is to use a generalized form of the circle equation. Instead of x² + y² = r², we'll use: |x|^n + |y|^n = r^n

When n = 2, this gives us a circle. As n increases, we get more "square-like" shapes.

Step 3: Adapt for N-sided polygons For an N-sided polygon, we'll use polar coordinates and create "sectors." The parametric equations become:

x(t) = r × sign(cos(Nt/2)) × |cos(Nt/2)|^(2/m) × cos(t) y(t) = r × sign(sin(Nt/2)) × |sin(Nt/2)|^(2/m) × sin(t)

Where:

• t goes from 0 to 2π (our parameter)
• N = number of sides
• r = radius/size of the shape
• m = our "smoothness parameter"

Step 4: Understanding the parameters

• When m = 2: You get a circle
• As m increases: Corners become sharper, approaching a true polygon
• N determines how many sides/corners your shape has

The Answer:

Final Formula: ``` x(t) = r × sign(cos(Nt/2)) × |cos(Nt/2)|^(2/m) × cos(t) y(t) = r × sign(sin(Nt/2)) × |sin(Nt/2)|^(2/m) × sin(t) ```

Parameters:

• t ∈ [0, 2π] (parameter for tracing the curve)
• N = number of sides (3 for triangle, 4 for square, etc.)
• r = size/radius of the shape
• m = smoothness control (2 = circle, larger values = sharper corners)

Memory Tip:

Remember "Smooth Superellipse": The Sign function handles the quadrants, the Special exponent (2/m) controls smoothness, and we're essentially "modulating" a circle with N-fold symmetry!

Think of it as taking a circle and "pinching" it at N equally-spaced points - the more you pinch (larger m), the more polygon-like it becomes! 🌟