When I needed to pass Calculus and Algebra during my Computer Science degree, I used the formula known to every Polish student called 'ZZZ' (Zakuć, Zdać, Zapomnieć) - Cram, Pass, Forget. It means memorizing just enough to pass the exam and then not caring if you'll remember any of it later.

Fast forward a few years into my web developer career and I needed to implement a mouseMove event on the SVG element. Every move I made with my mouse made the element's position shoot off into the void. I solved this issue using the transformation matrix.

The more I worked with SVGs and graphs, the more I realized that the math I tried to cram and forget shows up in everything that I do. From the coordinates to graph layouts, drag-and-drop to animations - we just don't call it 'math'.

This series attempts to show you that math. Each article takes one concept (that we had to cram in class) and shows exactly where it lives in real frontend code, with real examples I've built in production or in side projects. I don't want to show you mathematical proofs, I will explain a bit of theory with a demonstration how to use it when you are the one who has to position elements just the right way, resolve conflicts and overlaps or draw a line from one point to another.

Welcome to Know Your WebDev Math.

Part 1: Dynamic positioning with trigonometry

Introduction

I play a ton of DnD. And in my longest and favourite campaign (Curse of Strahd) the relations within the party and between the party and important NPCs (Ireena) are extremely important. We keep an Excel spreadsheet describing the current relationship status (from 'love' to 'hate' to 'it's complicated'). And I had an idea of creating an interactive diagram of those relations.

Requirements

1. Every character avatar and name needs to be visible.
2. Every character needs to be connected to every other character.
3. The color of the connection symbolizes the relationship status.
4. Hovering over a character's avatar should show description of what they think of others and what others think of them.
5. The number of characters is not fixed.

Analysis

Graph is the best structure to show elements and their relations, so I decided to implement my diagram in React Flow.
To ensure good readability of the graph I decided to use circular graph layout:

It means that each character avatar should be positioned on the edge of the circle.
Only one question remained: How to dynamically position elements in the circle?

Solution: Trigonometry

Theory: sine and cosine

This is a right-angled triangle.

(Bear with me for a moment, I know you know what a triangle is. All of this will matter in a second)

It has 3 sides: opposite, adjacent and hypotenuse. It also has an acute angle, let's call this angle α.

We can calculate sine of α - sin(α) - by dividing the length of the opposite side by the length of hypotenuse side: sin(α) = opposite / hypotenuse.

Same with cosine of α - cos(α) - we need to divide the length of the adjacent side by the length of the hypotenuse side: cos(α) = adjacent / hypotenuse.

Sine and cosine are ratios, which means they stay exactly the same for every specific angle value. The sine of 30 degree angle is always 0.5 - it doesn't matter if the opposite side of a triangle has a length of 2 or 200.

How is it going to help us?

We want to position elements on the edge of the circle in equal distances from each other - we need to calculate x and y position of the centre of each element.

Let's go back to our circular graph. If we draw the line from the centre of the graph to the centre of one of the nodes, we can create a right-angled triangle in which the lengths of adjacent and opposite sides are our x and y, respectively.

We need to find the lengths of sides in our triangle. And we can do that using sine and cosine.

Calculations

The easiest side is hypotenuse - it's the radius of our layout circle, let's call this variable radius. It can be arbitrary, just adjust it to the size of your nodes.
The center of the graph (point C) can also be arbitrary, let's assume variable c of type Point: type Point = {x: number; y: number}.

To calculate the value of x - adjuscent side of triangle - let's change the cosine equation:
adjacent = cos(α) * hypotenuse.
And to calculate the value of y - opposite side - let's change the sine equation the same way:
opposite = sin(α) * hypotenuse.

We can calculate the sine and cosine values for specific angle α using functions from JavaScript Math library: Math.sin and Math.cos. Those functions take radian angle value as parameter.

So how can we calculate radian value of α?

We know that our graph is going to have n elements. And radian value of 360 degree angle is 2π.

So our α is: α = 2π / n or in JS: const alpha = (2 * Math.PI) / n.

Now, we need to iterate over each node and calculate its position:

const updatedNodes = nodes.map((node, index) => {
        const angle = index * alpha;
        return {
          ...node,
          position: {
            x: c.x + radius * Math.cos(angle),
            y: c.y + radius * Math.sin(angle),
          },
        };
     });

Nodes above are graph nodes from React Flow, so if I'll set position property in the node object, React Flow will automatically place them for me.

Result

Behold, an example with Baldur's Gate 3 characters!

Additionally, you can adjust your radius variable based on the number of nodes in the graph to have fully dynamic graph, growing with the number of characters you want to have in your D&D (or DaggerHeart, or Call of Cthulhu or any other system) session.

I decided to start Know Your WebDev Math series with simple trigonometry, because sine and cosine are, for me, the symbols of 'that stupid math I won't ever use in my life' approach. Turns out, trigonometry was just waiting for me to build something worthy using it on.

If you play D&D or have a graph layout problem, drop it in the comments!

Next up: the tower of Babel of different coordinate systems, and how transformation matrix acts as a translator.