**Initiating Development: HTML5 Canvas**

I've started with a blank canvas and decided to use HTML5 Canvas for a polished, interactive application. My focus is on visuals; I'm leaning toward a modern, dark mode aesthetic for a sleek canvas UI. I want this application to be stunning, robust and performant.


**Refining Visuals & Controls**

I'm now thinking about specific visual enhancements and user controls. I envision glowing neon or crisp vector styling, and want particle effects on the bounce. I'm focusing on adding pause/resume, reset/restart, gravity, restitution sliders, and speed/spin toggles. I need to add clear/add balls buttons and display stats like FPS, active balls, reset button, and pause button. I'm aiming for a beautiful heptagon geometry with a glowing/pulsing outline, smoothly rotating at 72 degrees/sec.


**Implementing Physics Interactions**

I'm now fully immersed in the physics, focusing on the core mechanics. I'm excited about making realistic bounces with restitution and incorporating tangential friction for rotating walls. I'm working with 20 balls, each with its own position, velocity, and properties like radius, mass, rotation, and spin. I'm implementing gravity, friction, and detailed ball-ball collision detection/response including angular velocity transfer! I also want to deal with ball-wall collisions by determining the closest point on each wall segment.


**Analyzing Wall Interactions**

I'm now deep-diving into the ball-wall interaction physics. I'm focusing on calculating relative velocities, breaking them into normal and tangential components to handle bounce and friction, including rolling friction, and applying positional corrections to keep the balls from getting stuck. The plan is to impart spin on balls as they collide with rotating walls.
I'm also considering the interactive controls and layout for my single-file `index.html` file, remembering to avoid external resources.


**Refining: Resource Restrictions**

Okay, so I have to work within constraints! All resources must be internal to the `index.html` file, no external CDNs or network requests allowed. This means pure vanilla CSS and JS, which excites me. I am planning a dark background, system sans-serif font, and a glassmorphism style control panel. I must ensure the canvas is responsive and high-DPI scaled for clarity.



**Calculating Collision Vectors**

I'm now deep in collision detection! For each ball and wall segment, I'm calculating the closest point on the segment and the distance. If it's less than the ball's radius, I have a collision. I'm focusing on the normal vector calculation to make sure it always points from the wall to the ball inside the heptagon. I plan to use this information for an appropriate response.


**Defining Inward Normals**

I'm now determining how to calculate and define inward-pointing normals for each heptagon wall. I'm considering cross products or polygon containment tests to ensure normals are consistently directed towards the center. The goal is to derive the appropriate normal vectors for accurate collision responses. I am working through the relative velocity components now, for calculating the wall velocity at the contact point.



**Implementing Friction, Spin and Torque**

I'm working on the final steps of interaction! I'm now implementing tangential friction and torque. I'm calculating the tangential velocity component, and adjusting based on the wall's friction coefficient. This affects the ball's angular velocity - which means spin! I'm updating the ball's velocity and position, and handling the positional corrections.

I am also deep diving on ball-ball collisions: I calculate the distance vector, check if the distance is less than the min distance - and if so, calculate the normal, and adjust the separation based on each ball's mass.


**Calculating Ball-Ball Collisions**

I'm focusing now on the specifics of ball-ball collisions. With equal mass, I move each ball by half the separation distance along the normal. The relative velocity is crucial: it's the difference between the velocities of the two balls. I'm calculating the normal velocity component and applying restitution if it's negative. I calculate the impulse scalar, and update the velocities to implement tangential friction and transfer momentum and spin!

**Verifying the Code**

I'm now diving into the code verification process. My focus is to ensure the code functions as expected and to identify potential areas for optimization. I've begun reviewing the requirements, and have confirmed that the numbered balls are correctly implemented. I am using the ball ID mapping (0-19) to the specified numbers (1-20).

**Investigating Physics Bugs**

I'm currently focused on potential physics bugs and edge cases, specifically concerning ball spawning at the center of the heptagon. My primary concern is if balls, starting near the center at (cx, cy) with a radius of 240, will be trapped or fall outwards. Initial tests indicate a downward movement.


**Analyzing Wall Dynamics**

Now I'm diving deeper into wall velocity and normal orientation within `handleCollisions()`. It seems a sign error might exist with the vector pointing to center. I am carefully assessing the vector calculations to ensure normals point correctly inward, especially given the ball's position relative to the wall, determining if and how these calculations affect outward fall direction.


**Verifying Vector Directions**

I've confirmed that negating the normal components `nx` and `ny` correctly directs the vector inward, toward the heptagon's center. I'm now verifying the wall velocity calculation; everything looks right so far. I have confirmed that wall velocities are being calculated correctly and am checking it using the cross product approach and rotation matrices. I am cross-checking this mathematically and in canvas coordinates.


**Confirming Clockwise Rotation**

I'm now deeply engaged in confirming the clockwise rotation calculations within the canvas coordinate system. I'm rigorously testing the relationship between angular speed ($\omega$) and the velocity components ($vx$, $vy$) of points rotating around the heptagon's center. It seems my initial derivation had a sign error!


**Validating Coordinate System**

I've just confirmed the coordinate system and rotation calculations within the canvas. My recent tests validated the clockwise rotation formulas. Now, I'm considering enhancing the simulation with mouse interaction, specifically implementing click-and-drag functionality to move the balls! I think this could make the simulation much more interactive and delightful. I'll focus on the mousedown event to find the closest ball for dragging.