DesThree

Desmos is a wonderful user-friendly calculator for graphing in two dimensions. However, creating three-dimensional projects can be confusing at first, but more importantly they are slow due to essentially having to recompute meshes and projections on every frame.

Three.js is an excellent rendering library that simplifies much of the process of WebGL and web shaders.

The goal of this project is to unite the two:

• The user-friendliness of Desmos, including sliders, MathQuill, and more
• The power of WebGL to make fast 3D animations easily.

This project is unofficial, so it may break at any time due to changes in the Desmos code.

Installation

1. Install the TamperMonkey browser extension
2. Click on DesThree.user.js in the latest Release, then hit Install.

• ⚠️ Be sure you trust myself and the source code. Malicious userscripts can make unwanted access to your data (this one does not).

3. Open https://www.desmos.com/calculator/7kykpkx0ji to test it! You should see 9 blue-ish boxes moving up and down.
4. Updates should be handled automatically. If you want to manually check for updates, click Check for userscript updates on the Tampermonkey extension icon.

Please report any bugs here on the Github issues tracker or discuss on the Unofficial Desmos Discord in #programming.

Development

1. Clone the repository
2. Allow TamperMonkey access to file URLs
3. npm install
4. npm run dev
5. Load dist/DesThree-dev.user.js into TamperMonkey
6. The script should automatically update (except for metadata changes) because dist/DesThree-dev.user.js loads dist/DesThree.user.js through file URL

Use npm run lint or npm run lint-fix before a commit to adhere to Standard style. You may also want the packages linter-js-standard and atom-standard-formatter for convenience, though npm run lint-fix is more thorough than whatever atom-standard-formatter does.

To test a production build:

1. Run npm run build-tests and paste dist/DesThree-test.user.js into a new TamperMonkey script
2. Run npm run build and paste dist/DesThree.user.js into a TamperMonkey script
3. Make sure that these are the only two active userscripts
4. Run npm run test, which lints and opens https://www.desmos.com/calculator?testDesThree=0. If this doesn't work, open the URL manually.
5. Expected behavior is shown in the browser console. Follow the directions and click Pass to move on to each next test.

Periodically, update test definitions based on Chrome's code coverage tool.

To build once for production, use npm run build instead of npm run dev. The built script should be available in dist/DesThree.user.js, which can be pasted directly into TamperMonkey.

Other release details are available in RELEASE.md

Usage

Changelog

DesThree will automatically load in every graph you load with the userscript enabled, and at present, it completely covers the normal 2D graph.

To create a DesThree expression, insert it from the ➕ menu in the top-left as you might insert a folder or an image. You can also type @3 at the start of a normal expression to begin a DesThree expression; the icon to the left should switch to the shape of a cube.

If DesThree ever stops showing something or stops being responsive, save and reload the page. If it still doesn't work, it's probably the fault of your expressions.

If you ever can't type something (e.g. PointLight automatically replaces int with an integral), type it in some other tab/program and copy-paste into Desmos.

Example 1: Setting Camera Position

The simplest expression you can write simply sets the position of the camera as (x,y,z)=(1,2,5) (https://www.desmos.com/calculator/i8min5dnfo):

pos = (1,2,5)
PerspectiveCamera(pos)

• You can't see anything because we didn't place anything in the scene.
• All functions start with capital letters
• Variable names must start with a letter but can have letters, digits, or underscores elsewhere:
  • Valid: cameraPos, camera_pos, cube_1, thr33js
  • Invalid: 12cube, _moo
• The y-axis points up by default.
• The camera points to (0,0,0) by default.

Example 2: Showing a Cube

Showing a cube may be more complicated than what you expect. The process goes as follows:

• Define a cube geometry. This holds the shape of the cube but cannot be shown by itself.
• Define the material to be used.
• Create a mesh, which combines the material and geometry into one object. Meshes default to being placed at (0,0,0).
• Show the mesh

These steps can be done as follows (https://www.desmos.com/calculator/6t1l6vhfiy):

pos = (1,2,5)
PerspectiveCamera(pos)
cubeGeometry = Box(1,1,1)
material = NormalMaterial()
mesh = Mesh(cubeGeometry, material)
Show(mesh)

Alternatively, you can place them all in one expression (https://www.desmos.com/calculator/0cjxlfabx5):

PerspectiveCamera((1,2,5))
Show(Mesh(Box(1,1,1), NormalMaterial()))

• Be careful: PerspectiveCamera seems to have double parentheses, but the inner ones are for the point (1,2,5)—only the outer ones are for the function call.
• Box takes three arguments: width, height, length. To make a cube, we use the same value for all three
• NormalMaterial takes no arguments. Example 5 will go over the different kinds of materials
• Mesh simply takes the geometry and material and arguments.
• Show takes the mesh and shows it. This does not occur automatically because we sometimes want to store a mesh in a variable and not show it until after some transformations

Example 3: Moving Box Position

A great part about using Desmos is that we can use sliders and math in the expressions.

To see an example, we will move the box around in 3D. We use the Position function to create a new mesh that corresponds to the mesh translated (https://www.desmos.com/calculator/pygwi1lnpi):

PerspectiveCamera((1,2,5))
cubeGeometry = Box(1,1,1)
material = NormalMaterial()
mesh = Mesh(cubeGeometry, material)
x_0 = 0
y_0 = 0
z_0 = 0
meshp = Position(mesh, (x_0, y_0, z_0))
Show(meshp)

Alternatively, we can specify position directly in the Mesh function (https://www.desmos.com/calculator/ir00nvxzxs):

meshp = Mesh(cubeGeometry, material, (x_0, y_0, z_0))
Show(meshp)

• Desmos generates sliders for each position variable, which we can use in the function. Try dragging the sliders!
• Note that we are now showing meshp, the mesh correctly positioned in 3D.

Example 4: Moving Camera Position

To improve our future viewing, we will orbit the camera using equations of spherical coordinates.

This simply involves a few more equations instead of placing the camera at a fixed position (https://www.desmos.com/calculator/3cy48fxofl):

r_c = 5
theta_c = 0
phi_c = 0
pos = (r_c·cos(phi_c)cos(theta_c), r_c·sin(phi_c), r_c·cos(phi_c)sin(theta_c))
PerspectiveCamera(pos)
cubeGeometry = Box(1,1,1)
material = NormalMaterial()
mesh = Mesh(cubeGeometry, material)
x_0 = 0
y_0 = 0
z_0 = 0
meshp = Position(mesh, (x_0, y_0, z_0))
Show(meshp)

Example 5: Materials and Lights

Remember how we're using NormalMaterial above? That is the simplest material (and the default if none is given to the Mesh function): there are no arguments, and we don't have to deal with lights. Each face is simply colored based on its normal vector.

Two other materials exist that are always visible regardless of lights:

• DepthMaterial: also takes no arguments
• BasicMaterial: takes a color as argument and always shows that color regardless of lighting

Other materials exist for shading meshes, and these handle lights:

• LambertMaterial
• PhongMaterial
• ToonMaterial

Each of these take color as their only argument.

Use them as follows:

material = LambertMaterial(RGB(10, 80, 180))
mesh = Mesh(geometry, material)
show(mesh)

With these last few materials, you need to add lights if you want to see anything.

Let's add a single point light at (-7, 6, 2) with an intensity (brightness) of 1 (https://www.desmos.com/calculator/oznclpvvns):

Show(Position(PointLight(1), (-7, 6, 2)))

(Alternatively, you can use Show(PointLight(1, RGB(255,255,255), (-7,6,2))), which avoids the Position function).

Looks good right? Yes, but try rotating to the back side (https://www.desmos.com/calculator/imbwxpqh1a). Ugh, that's too dark.

One solution would be to add more point lights so nothing can be dark, but that would get confusing and not look good. Instead, let's add an ambient light, which adds the intensity to every face (https://www.desmos.com/calculator/lsnje9ira5):

Show(AmbientLight(0.3))

• Ambient lights have no position
• An ambient light fakes light reflecting off of other surfaces (wherever we look, there's normally a wall or ground reflecting light onto the back side of objects we look at; this is simulated by ambient light)
• In almost all cases, you want only one ambient light
• You normally want to stick with an intensity between 0.01 (dark faces can be almost completely black—good for moody scenes) to 0.5 (all faces are easily visible)

Example 6: Lists

Anywhere a number can be accepted, a list of values can be accepted instead.

Let's make a sphere with radius 1 and place it at x-coordinates [-10, -5, 0, 5, 10] (https://www.desmos.com/calculator/j4yf8gqofe):

geometry = Sphere(1)
mesh = Mesh(geometry, material)
meshp = Position(mesh, ([-10,-5,...,10], 0, 0))
Show(meshp)

Just like in vanilla Desmos, if several lists are passed to a function, a list is produced from applying the function to corresponding terms with the output length being the minimum of the lengths of the input lists (https://www.desmos.com/calculator/ebdz3ges1f):

L_x = [-10,-5,...,10]
mesh = Mesh(geometry, material)
meshp = Position(mesh, L_x, (0.05*L_x^2, 0))
Show(meshp)

You can even make a list of geometries with different parameters (https://www.desmos.com/calculator/zdy9szxjzy)

L_x = [-12.5,-7.5,...,12.5]
geometry = Sphere(0.02*L_x^2+0.5)
mesh = Mesh(geometry, material)
meshp = Position(mesh, L_x, (0.02*L_x^2, 0))
Show(meshp)

Go crazy, and have fun!

Function Reference

Functions that return a Geometry from numeric arguments

Icosahedron, Dodecahedron, Octahedron, Tetrahedron

Property	Type	Default	Description
radius	number		Radius of circumscribing sphere
detail	number	0	If detail > 0, the polyhedron becomes more rounded towards a sphere

Sphere

Property	Type	Default	Description
radius	number		Radius of circumscribing sphere
widthSegments	number	16	Greater = smoother
heightSegments	number	12	Greater = smoother

Torus

Property	Type	Default	Description
radius	number		Radius from center of torus to center of tube in cross section
tube	number		Radius of tube
radialSegments	number	16	Greater = smoother
tubularSegments	number	12	Greater = smoother
arc	number	2π	Arc angle

TorusKnot

Property	Type	Default	Description
radius	number
tube	number		Radius of tube
radialSegments	number	8	Greater = smoother
tubularSegments	number	64	Greater = smoother
p	number	2	How many times the geometry winds around its axis of rotational symmetry
q	number	3	How many times the geometry winds around a circle in the interior of the torus

Cone

Property	Type	Default	Description
radius	number
height	number
radialSegments	number	16	Greater = smoother
heightSegments	number	1

Frustum

Property	Type	Default	Description
radiusTop	number
radiusBottom	number
height	number
radialSegments	number	16	Greater = smoother
heightSegments	number	1

Cylinder

Just a convenience to avoid having to pass the same value for radiusTop and radiusBottom in Frustum.

Property	Type	Default	Description
radius	number
height	number
radialSegments	number	16	Greater = smoother
heightSegments	number	1

Disc

A filled circle (flat). Lowering the value of the segments argument would make a polygon.

Property	Type	Default	Description
radius	number
thetaStart	number	0
thetaLength	number	2π	Disc(1, 0, π) is a semi-circle
segments	number	16	Disc(1, 0, 2π, 6) is a hexagon

Ring

Also known as an annulus

Property	Type	Default	Description
innerRadius	number
outerRadius	number
thetaStart	number	0
thetaLength	number	2π
thetaSegments	number	16
phiSegments	number	1

Plane

A rectangular plane of finite size.

Property	Type	Default	Description
width	number
height	number
widthSegments	number	1
heightSegments	number	1

Functions that return a Geometry using a list of points

Shape

Connects 2D points into a flat shape, assuming their y coordinate is 0.

Property	Type	Default	Description
points	list of Vector2s
divisions	number	1	number of subdivisions to take of a spline passing through those points. If divisions = 1, then this a polygon between those points

Extrude

Connects 2D points into a polygon and extrudes to a specified depth

Property	Type	Default	Description
points	list of Vector2
divisions	number	1	number of subdivisions to take of a spline passing through those points. If divisions = 1, then this a polygon between those points
depth	number	10	distance to extrude

Lathe

May also be known as "revolve."

Resolves (x,y) points around the y-axis. The x values are automatically absolute-valued.

Property	Type	Default	Description
points	list of Vector2s
segments	number	12	greater = smoother
phiStart	number	0
phiLength	number	2π

BufferGeometry

Creates a geometry directly from a list of faces. Most powerful method for creating a geometry.

Property	Type	Default	Description
faces	list of Face3s

Functions that return a Material

BasicMaterial

Property	Type	Default	Description
color	Color	white

Always colored as the given color, ignoring lighting.

NormalMaterial

No arguments. Each face is colored based on its normal vector.

DepthMaterial

No arguments. Colors each point in gray-scale based on its depth from the camera. This is not very useful for general rendering and usually shows up mostly black unless the object is very close to the camera.

LambertMaterial

Property	Type	Default	Description
color	Color	white

This uses Gourand shading, which calculates shading for each vertex and does linear interpolation to find pixel shading. As such, it only tends to look best for low-poly objects like cubes/boxes.

PhongMaterial

Property	Type	Default	Description
color	Color	white

This uses Phong shading, which calculates shading on each pixel and additionally includes specular highlights (shiny surfaces). This tends to look better than MeshLambertMaterial for rounded objects like spheres but is slower.

ToonMaterial

Property	Type	Default	Description
color	Color	white

Looks like a cartoon drawing.

Functions that return an Object3D

Mesh

Create an object from a geometry and material.

Property	Type	Default	Description
geometry	Geometry
material	Material	NormalMaterial()
position	Vector3	(0,0,0)

SquareGrid

Draw a square grid of the given size

Property	Type	Default	Description
size	number		width, in world units
divisions	number	10	greater = more grid lines
center lines color	Color	white	only applies if divisions is even
grid color	Color	gray

PolarGrid

Draw a polar grid of the given size. The two colors alternate.

Property	Type	Default	Description
radius	number
radials	number	16	number of radial lines to draw
circles	number	6	number of circles to draw
divisions	number	64
color1	Color	white
color2	Color	white

ArrowHelper

Draw an arrow from tail with components given in vector.

Property	Type	Default	Description
tail	Vector3
vector	Vector3
color	Color	white
headLengthScale	number	0.2	length of the arrow head, as a multiplier of the length of the vector
headWidthScale	number	0.4	diameter of the arrow head, as a multiplier of the length of the arrow head

AxesHelper

Draws standard x, y, z axes.

Property	Type	Default	Description
size	number	1

PointLight

Property	Type	Default	Description
intensity	number	1	Brightness
color	Color	white
position	Vector3	(0,0,0)

AmbientLight

Property	Type	Default	Description
intensity	number	1	Brightness
color	Color	white

HemisphereLight

Like AmbientLight, but the color changes from sky to ground

Property	Type	Default	Description
intensity	number	1	Brightness
sky color	Color	white
ground color	Color	white
direction	Vector3	down = (0, -1, 0)	direction vector from sky to ground

DirectionalLight

Property	Type	Default	Description
intensity	number	1	Brightness
color	Color	white
direction	Vector3	down = (0, -1, 0)	direction of travel of light rays

SpotLight

Like a PointLight except it emits light only within a cone from its position pointing towards a target. The angle and other settings of the cone can be controlled

Property	Type	Default	Description
position	Vector3
target	Vector3	(0,0,0)
intensity	number	1	Brightness
color	Color	white
angle	number	60° = π/3
penumbra	number	0	Takes values from 0 to 1. Percent of attenuation due to the penumbra effect (outer lighter part of the shadow)
distanceLimit	number	0	Maximum distance of light (dims linearly). Set to 0 to allow infinite distance
decay	number	2	Set to 2 to follow inverse-square low

Position

Property	Type	Default	Description
object	Object3D		Geometry not accepted
position	Vector3		Vector by which to translate the object.

Functions that directly affect the scene

PerspectiveCamera

Property	Type	Default	Description
position	Vector3		Position of the camera
lookAt	Vector3	(0,0,0)	Position of the point the camera looks at
fov	number	75	field of view (degrees, planned: Change treatment based on radian mode of calculator)
near	number	0.1	near clipping plane of camera frustum
far	number	1000	far clipping plane of camera frustum

Show

Property	Type	Default	Description
object	Object3D

LinearFog

Property	Type	Default	Description
near	number
far	number
color	Color

FogExp

Property	Type	Default	Description
density	number
color	Color

Functions that return a Color

RGB

Property	Type	Default	Description
r	number		Red component, 0 to 255
g	number		Green component, 0 to 255
b	number		Blue component, 0 to 255

Miscellaneous

Vector2

Please note that Desmos points (declared in vanilla Desmos expressions) are not currently supported.

Property	Type	Default	Description
x	number
y	number

Vector3

Property	Type	Default	Description
x	number
y	number
z	number

Face

A collection of three vertices, used as an argument for BufferGeometry. Be careful with the order of the three vertices (clockwise or counterclockwise) because that affects computed face normals, which affects lighting

Property	Type	Default	Description
vertex a	Vector3
vertex b	Vector3
vertex c	Vector3