Deformation Functions

These functions are available in both the equation inputs and the normal deformation shader. The variables gx, gy, gz refer to vertex position, and gu, gv to parametric coordinates.

Surface deformation

Function	Formula	Description
m() m(c) m(cx,cy,cz)	cos(cx·x) × cos(cy·y) × cos(cz·z)	Multiplicative cosine – creates a 3D grid-like pattern
o() o(c) o(cx,cy,cz)	cos(cx·x) + cos(cy·y) + cos(cz·z)	Additive cosine – creates gyroid / Schwarz-like shapes
f(nc, np) f(nc, npx, npy, npz) f(ncx,npx,ncy,npy,ncz,npz)	cos(nc·x + np) × cos(nc·y + np) × cos(nc·z + np)	Phase-shifted cosine product – like m() with phase offset
a() a(n) a(nU, nV)	cos(nU·u) × sin(nV·v)	Angular UV deformation – pattern tied to parametric coords
b() b(c) b(cx,cy,cz)	length(cos(cx·x), cos(cy·y), cos(cz·z))	Euclidean length of cosines
ce(c)	cos(ec|x|) × cos(ec|y|) × cos(ec|z|)	Exponential cosine – rapid frequency increase outward
se(c)	sin(ec|x|) × sin(ec|y|) × sin(ec|z|)	Exponential sine – same as ce() but with sin

Fragment shader only (color shader)

Function	Formula	Description
hc(x,y,z) hc(vec3 p) hc(x,y,z,c)	length(cos(x), cos(y), cos(z))	Hypot of cosines – for color patterns

Math utilities

Function	Formula	Description
c(v)	cos(v)	Short cosine alias
s(v)	sin(v)	Short sine alias
ca(v)	0.5 × cos(v) + 0.5	Cosine normalized to [0, 1]
sa(v)	0.5 × sin(v) + 0.5	Sine normalized to [0, 1]
h(x,y) / h(x,y,z)	length(vec2/3(...))	Euclidean distance (hypot)
q(a, b, t)	mix(a, b, t)	Linear interpolation
r(e0, e1, x)	smoothstep(e0, e1, x)	Smooth Hermite interpolation
g(edge, x)	step(edge, x)	Step function (0 or 1)
cpow(v, p)	sign(v) × |v|p	Signed power (preserves sign)

Vector operations

Function	Formula	Description
cr(x1,y1,z1, x2,y2,z2)	cross(v1, v2)	Cross product (returns vec3)
crl(x1,y1,z1, x2,y2,z2)	length(cross(v1, v2))	Cross product length
cc(x1,y1, x2,y2)	dot(v1, v2)	Dot product (2D or 3D)

Writing Shortcuts

The equation parser automatically expands shorthand notations. This lets you write compact formulas that get expanded before compilation.

Trigonometric shortcuts

You write	Expands to	Description
cu	cos(u)	c + variable = cos(variable)
cv	cos(v)
su	sin(u)	s + variable = sin(variable)
sv	sin(v)
cx	cos(x)	Also works with x, y, z, t
sy	sin(y)
ct	cos(t)

Implicit multiplication

Variables and parentheses are automatically multiplied when juxtaposed:

You write	Expands to
2u	2*u
uv	u*v
3pi	3*pi
AB	A*B
)(	)*(

Other shortcuts

You write	Expands to	Note
R	h(x,y,z)	Radius – distance from origin
l	log	Natural logarithm
²	**2	Square
³	**3	Cube
ù	*3mct	Time multiplier shortcut

Keyboard Shortcuts

Camera & View

Key	Action
+	Zoom in
-	Zoom out
Shift + H	Center camera on origin
Shift + V	Align camera to axis view
Shift + B	Toggle wireframe
Alt + +	Increase auto-rotation speed
Alt + -	Decrease auto-rotation speed

UV Parameter Animation

Key	Action
7 / 8	Animate U − / + (large step)
0 / 1	Animate U − / + (small step)
4 / 5	Animate V − / + (large step)
6 / 9	Animate V − / + (small step)
u	Increase mesh resolution
j	Decrease mesh resolution

Time & Animation

Key	Action
a	Toggle pause / play
f	Double time speed
q	Halve time speed

Colors & Display

Key	Action
h	Randomize colors
"	Special randomize (high variation)
*	Initialize color UI

Other

Key	Action
$	Generate random surface
p	Open import dialog
'	Toggle UV ↔ XY mode
; / ,	Switch writing type − / +
<	Toggle form suite mode
Shift + Q	Copy X equation to Y and Z

Shader Editor

Key	Action
Ctrl + S	Compile shader
Ctrl + D	Duplicate line

Coordinate Systems

SURFACE supports three coordinate systems. The equations you write in X, Y, Z are interpreted differently depending on the active system.

Cartesian (default)

Axis	Equation
X	x = f(u, v)
Y	y = f(u, v)
Z	z = f(u, v)

Direct mapping – each axis is defined independently as a function of u and v.

Spherical

Input	Role	Conversion
X → r	Radius	x = r × sin(θ) × cos(φ)
Y → θ	Polar angle	y = r × sin(θ) × sin(φ)
Z → φ	Azimuthal angle	z = r × cos(θ)

Best for spheres, globes, and closed shells.

Cylindrical

Input	Role	Conversion
X → r	Radius	x = r × cos(θ)
Y → θ	Angle	y = r × sin(θ)
Z → z	Height	z = z

Best for tubes, spirals, and surfaces of revolution.

Shader Editor

SURFACE includes two GLSL shader editors that let you customize the visual output in real time.

Fragment (Color) Shader

Controls the color of each pixel on the mesh surface. The editable zone starts after vec3 col = meshBg; and ends before // __FOOTER_START__.

• col – the output color (vec3 RGB)
• vPosition – vertex position in world space
• vNormal – surface normal
• vUV – parametric UV coordinates
• uTime – elapsed time
• meshBg – default mesh background color

Normal Deformation Shader

Displaces vertices along their normals. The editable zone is between float result = 0.0; and return result;.

• result – the displacement amount (float)
• x, y, z – vertex position
• nx, ny, nz – normal direction
• All deformation functions (m, o, f, a, b, etc.) are available

Shader management

• Select – switch between saved shaders via the dropdown
• New – create a new shader (named via first-line comment // My Shader)
• Save – save current shader to browser localStorage
• Delete – remove current shader
• Export / Import – download or upload all shaders as a .js file
• Opt 1, 2, 3 – toggle shader options (checkboxes)

Symmetry & Transformations

Symmetrize

Repeats the mesh along each axis to create symmetric patterns.

Control	Description
X, Y, Z	Number of repetitions along each axis (1 = no repetition, up to 48)
Angle	Rotation angle between repetitions
Checkerboard	Alternating pattern of repetitions (0 = off)
Scale	Scale factor for normal deformation
S order	Order in which X, Y, Z symmetries are applied (click to cycle)
S ADD / MUL	Additive or multiplicative symmetry blending

Transformations

Transform	Description
Scaling X/Y/Z	Scale the mesh along each axis
Rotation X/Y/Z	Rotate the mesh (in radians)
Position X/Y/Z	Translate the mesh
Center Symmetry X/Y/Z	Move the symmetry center point

Blender

Blends two surface components together:

• U (X/Y/Z) – first blend factor per axis
• O (X/Y/Z) – second blend factor per axis

Waves

The Waves panel adds normal-based wave deformation with linked sliders:

• Norm X/Y/Z – wave amplitude per axis
• n X/Y/Z – wave frequency coefficient per axis

Colors

Control	Description
Add R/G/B	Add a constant color offset to the shader output
Tint	Overall color intensity multiplier (0–2)

Import / Export

Export formats

Format	Description	Best for
JSON	Full parametric surface data (equations, parameters, colors)	Reloading in SURFACE
OBJ	Standard 3D mesh (vertices + faces)	Blender, Maya, 3D printing prep
STL	Triangulated mesh	3D printing

Import

• JSON – restores a previously exported SURFACE scene
• OBJ – loads an external 3D mesh (replaces current surface)
• Example JSON surfaces are available in the import dialog dropdown

Shader files

• Export shaders – downloads all fragment/normal shaders as a .js file
• Import shaders – load shaders from a .js file (replace or append)