Tangent Line

What is a Tangent Line?

A tangentLine draws the tangent to a parametric curve at a given parameter value $t$ . It creates a point on the curve and a line segment aligned with the derivative direction. Useful for:

Visualizing the derivative of a curve
Showing instantaneous direction of motion
Combining with curvePoint for interactive tangent exploration

The tangent direction is computed numerically, so you don't need to provide a derivative function.

Basic Usage

import { tangentLine } from "uzay";

const f = (t: number) => vec3(Math.cos(t), Math.sin(t), t * 0.3);

scene.create("parametricfunction3d", {
  f,
  tStart: 0,
  tEnd: Math.PI * 4,
  samples: 200,
  color: "dodgerblue",
});

const tl = tangentLine(scene, {
  f,
  t: 1.0,
  color: "yellow",
  length: 2,
});

This draws a yellow tangent line of length 2 at $t = 1$ on the helix, centered on the curve point.

Reactive Parameter

Pass an atom for t to make the tangent update dynamically:

const tAtom = scene.atom(0);

const tl = tangentLine(scene, {
  f,
  t: tAtom,
  color: "yellow",
});

// Move the tangent along the curve
tAtom.set(Math.PI);

Examples

Interactive Tangent with Curve Point

The most common pattern: combine with curvePoint so dragging the point moves the tangent:

const f = (t: number) => vec3(Math.cos(t), Math.sin(t), t * 0.3);

const { t } = curvePoint(scene, {
  f,
  tStart: 0,
  tEnd: Math.PI * 4,
  initialT: 1.0,
  color: "gold",
});

const tl = tangentLine(scene, {
  f,
  t: t,
  color: "tomato",
  length: 3,
  showPoint: false,
});

Reading the Tangent Direction

The returned tangent atom gives you the raw derivative vector (not normalized), which you can use for further computation:

const tl = tangentLine(scene, { f, t: tAtom });

// Subscribe to the tangent direction
tl.tangent.sub((dir) => {
  console.log("speed:", Vec3.length(dir));
});

Hiding the Point

If you're providing your own point (e.g. from curvePoint), you can hide the one created by the tangent line:

const tl = tangentLine(scene, {
  f,
  t: t,
  showPoint: false,
});

Customizing

The returned items are regular Point3D and Line3D:

const tl = tangentLine(scene, { f, t: 0.5 });

tl.point.radius.set(4);
tl.line.thickness.set(2);

API Reference

Options

Property	Type	Default	Description
`f`	`(t: number) => Vec3`	(required)	The parametric function defining the curve
`t`	`number`	(required)	Parameter value where the tangent is computed
`length`	`number`	`2`	Total length of the tangent line segment
`color`	`string`	`"yellow"`	CSS color string
`showPoint`	`boolean`	`true`	Whether to show the point on the curve

All options accept either a plain value or an atom.

Returned Handle

Field	Type	Description
`point`	`Point3D`	The point item on the curve
`line`	`Line3D`	The tangent line segment
`tangent`	`BoundAtom<Vec3>`	The tangent vector (derivative) at the current $t$
`dispose`	`() => void`	Removes all items from the scene

Tangent Line

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