# Introduction

Euclid is a Swift library for creating and manipulating 3D geometry using techniques such as extruding or "lathing" 2D paths to create solid 3D shapes, and CSG (Constructive Solid Geometry) to combine or subtract those shapes from one another.

Euclid is the underlying implementation for the ShapeScript App. Anything you can build in ShapeScript can be replicated programmatically in Swift using this library.

If you would like to support the development of Euclid, please consider buying a copy of ShapeScript (the app itself is free, but there is an in-app purchase to unlock some features).

**Note:** Euclid is a fairly complex piece of code, at a fairly early stage of development. You should expect some bugs and breaking changes over the first few releases, and the documentation is a little sparse. Please report any issues you encounter, and I will do my best to fix them.

# Installation

Euclid is packaged as a dynamic framework that you can import into your Xcode project. You can install this manually, or by using CocoaPods, Carthage, or Swift Package Manager.

**Note:** Euclid requires Xcode 10+ to build, and runs on iOS 10+ or macOS 10.12+.

To install Euclid using CocoaPods, add the following to your Podfile:

```
pod 'Euclid', '~> 0.1'
```

To install using Carthage, add this to your Cartfile:

```
github "nicklockwood/Euclid" ~> 0.1
```

To install using Swift Package Manage, add this to the `dependencies:`

section in your Package.swift file:

```
.package(url: "https://github.com/nicklockwood/Euclid.git", .upToNextMinor(from: "0.1.0")),
```

# Contributing

Feel free to open an issue in Github if you have questions about how to use the library, or think you may have found a bug.

If you wish to contribute improvements to the documentation or the code itself, that's great! But please read the CONTRIBUTING.md file before submitting a pull request.

# Types

The key types defined in the Euclid library are described below:

## Vector

The `Vector`

type represents a position or distance in 3D space.

There is no 2D vector type in Euclid, but when working with primarily 2D shapes (such as `Path`

s) you can omit the Z coordinate when constructing a `Vector`

and it will default to zero.

## Vertex

The `Vertex`

type is used to construct `Polygon`

s to form a `Mesh`

. Each `Vertex`

has the following `Vector`

properties:

`position`

- the`Vertex`

's location in 3D space`normal`

- the surface normal of a`Mesh`

at the vertex's position (used for lighting)`texcoord`

- a 2D texture coordinate used for texture mapping the`Polygon`

**Note:** The position of each `Vertex`

is automatically *quantized* (rounded to the nearest point in a very fine grid) in order to avoid the creation of very tiny polygons, or hairline cracks in surfaces. For that reason, to avoid accumulating rounding errors you should generally avoid applying multiple `Transform`

s to the same geometry in sequence.

## Polygon

The `Polygon`

type represents an arbitrary polygon in 3D space. `Polygon`

s must have three or more `Vertex`

es, and those vertices must all lie on the same plane. The edge of the `Polygon`

can be either convex or concave, but not self-intersecting.

The `Polygon`

constructor does some basic checks to try to prevent invalid `Polygon`

s from being constructed accidentally, but if you are sufficiently determined, you can probably still create something that will display incorrectly (or crash).

## Mesh

The `Mesh`

type represents a collection of `Polygon`

s that form a solid shape. A `Mesh`

surface can be convex or concave, and can have zero volume (e.g. a flat shape like a square) but should not contain holes or exposed backfaces, otherwise the result of CSG operations on the `Mesh`

will be undefined.

## Bounds

The `Bounds`

type represents an axis-aligned bounding box for a 3D shape or collection of shapes, such as `Polygon`

s or `Mesh`

es.

## Plane

The `Plane`

type represents an infinite plane in 3D space. It is defined by a surface normal `Vector`

and a `w`

value that indicates the distance of the center of the `Plane`

from the world origin.

## PathPoint

A `PathPoint`

is a control point along a path. `PathPoint`

s have a position `Vector`

, but no normal. Instead, the `isCurved`

property is used to indicate if a point is sharp or smooth, allowing the normal to be inferred automatically when required.

## Path

A `Path`

is a sequence of `PathPoint`

s representing a line or curve formed from straight segments joined end-to-end. A `Path`

can be either open (a *polyline*) or closed (a *polygon*), but should not be self-intersecting or otherwise degenerate.

`Path`

s may be formed from multiple subpaths, which can be accessed via the `subpaths`

property.

A closed, flat `Path`

without nested subpaths can be converted into a `Polygon`

, but it can also be used for other purposes, such as defining a cross-section or profile of a 3D shape.

`Path`

s are typically 2-dimensional, but because `PathPoint`

positions have a Z coordinate, they are not *required* to be. Even a flat `Path`

(where all points lie on the same plane) can be translated or rotated so that its points do not necessarily lie on the *XY* plane.

## Rotation

A `Rotation`

represents an orientation or rotation in 3D space. Internally, `Rotation`

is stored as a 3x3 matrix, but that's an implementation detail that may change in future. `Rotation`

s can be converted to and from an axis vector and angle, or a set of 3 Euler angles (pitch, yaw and roll).

## Transform

A `Transform`

combines a `Rotation`

with a pair of `Vector`

s defining the position and scale.

`Transform`

s are a convenient way to store and manipulate the location, orientation and size of `Mesh`

es without directly modifying the vertex positions (which can be problematic due to the buildup of rounding errors, as mentioned earlier).

# Geometry

You can create a `Mesh`

in Euclid by manually creating an array of `Polygon`

s, but that's pretty tedious. Euclid offers a number of helper methods to quickly create complex geometry:

## Primitives

The simplest way to create a `Mesh`

is to start with an existing primitive, such as a cube or sphere. The following primitive types are available in Euclid, and are defined as static constructor methods on the `Mesh`

type:

`cube`

- A cubic`Mesh`

(or cuboid, if you specify different values for the width, height and/or depth).`sphere`

- A spherical`Mesh`

.`cylinder`

- A cylindrical`Mesh`

.`cone`

- A conical`Mesh`

.

All `Mesh`

es are made of flat polygons, and since true curves cannot be represented using straight edges, the `sphere`

, `cylinder`

and `cone`

primitives are actually just approximations. You can control the quality of these approximations by using the `slices`

and/or `stacks`

arguments to configure the level of detail used.

In addition to the 3D `Mesh`

primitives listed, there are also 2D `Path`

primitives. These are implemented as static constructor methods on the `Path`

type instead of `Mesh`

:

`ellipse`

- A closed, elliptical`Path`

.`circle`

- A closed, circular`Path`

.`rectangle`

- A closed, rectangular`Path`

.`square`

- Same as`rectangle`

, but with equal width and height.

## Builders

Geometric primitives are all very well, but there is a limit to what you can create by combining spheres, cubes, etc. As an intermediate step between the extremes of using predefined primitives or individually positioning polygons, you can use *builders*.

Builders create a 3D `Mesh`

from a (typically) 2D `Path`

. The following builders are defined as static constructor functions on the `Mesh`

type:

`fill`

- This builder fills a single`Path`

to create a pair of`Polygon`

s (front and back faces).`lathe`

- This builder takes a 2D`Path`

and rotates it around the Y-axis to create a rotationally symmetrical`Mesh`

. This is an easy way to create complex shapes like candlesticks, chess pieces, rocket ships, etc.`extrude`

- This builder fills a`Path`

and extrudes it along its axis. This turns a circular path into a cylinder, or a square into a cube etc.`loft`

- This builder is similar to`extrude`

, but takes multiple`Path`

s and joins them. The`Path`

s do not need to be the same shape, but must all have the same number of points and subpaths. To work correctly, the`Path`

s must be pre-positioned in 3D space so they do not all lie on the same plane.

## Curves

Builders are a powerful tool for creating interesting `Mesh`

es from `Path`

s, but what about creating interesting `Path`

s in the first place?

Creating polygonal `Path`

s by specifying points individually is straightforward, but creating *curves* that way is tedious. That's where *Beziers* come in. Beziers allow you to specify complex curves using just a few control points. Euclid exposes this feature via the `curve`

constructor method.

The `curve`

method takes an array of `PathPoint`

s and a `detail`

argument. Normally, the `isCurved`

property of a `PathPoint`

is used to calculate surface normals (for lighting purposes), but with the `curve`

method it actually affects the shape of the `Path`

.

Regular (non-curved) `PathPoint`

s create sharp corners in the `Path`

as normal, but curved ones are treated as off-curve Bezier control points. The `detail`

argument of the `curve`

method controls how many straight line segments are used to approximate the curve.

The `curve`

method uses second-order (quadratic) Bezier curves, where each curve has two on-curve end points and a single off-curve control point. If two curved `PathPoint`

s are used in sequence then an on-curve point is interpolated between them. It is therefore possible to create curves entirely out of curved (off-curve) control points.

This approach to curve generation is based on the popular TrueType (TTF) font system, and provides a good balance between simplicity and flexibility.

For more complex curves, on macOS and iOS you can create Euclid `Path`

s from a Core Graphics `CGPath`

by using the `CGPath.paths()`

extension method. `CGPath`

supports cubic bezier curves as well as quadratic, and has handy constructors for rounded rectangles and other shapes.

## CSG

CSG (Constructive Solid Geometry) is another powerful tool for creating intricate geometry. CSG allows you to perform boolean operations (logical AND, OR, etc) on solid shapes. The following CSG operations are defined as methods on the `Mesh`

type:

`subtract`

- Subtracts the volume of one`Mesh`

from another.`xor`

- Produces a shape representing the non-overlapping parts of the input`Mesh`

es (this is useful for rendering text glyphs).`union`

- Combines two intersecting`Mesh`

es, removing internal faces and leaving only the outer shell around both shapes (logical OR).`intersection`

- Returns a single`Mesh`

representing the common volume of two intersecting`Mesh`

es (logical AND).`stencil`

- This effectively "paints" part of one`Mesh`

with the material from another.

All CSG operations require `Mesh`

es that are "watertight", i.e. that have no holes in their surface. Using them on `Mesh`

es that are not sealed may result in unexpected results.

## Text

On macOS and iOS you can make use of Euclid's Core Text integration to create 2D or 3D extruded text.

The `Path.text()`

method produces an array of 2D `Path`

s representing the contours of each glyph in an `AttributedString`

, which can then be used with the `fill`

or `extrude`

builder methods to create solid text.

Alternatively, the `Mesh(text:)`

constructor directly produces an extruded 3D text model from a `String`

or `AttributedString`

.

Each glyph in the input string maps to a single `Path`

in the result, but these `Path`

s may contain nested subpaths. Glyphs formed from multiple subpaths will be filled using the even-odd rule (equivalent to an `xor`

between the individually filled or extruded subpaths).

# Rendering

It's all very well creating interesting 3D geometry, but you probably want to actually *do something* with it.

Most of the Euclid library is completely self-contained, with no dependencies on any particular rendering technology or framework. However, when running on iOS or macOS you can take advantage of Euclid's built-in SceneKit integration. This is demonstrated in the Example app included with the project.

SceneKit is Apple's high-level 3D engine, which can use either OpenGL or Metal for rendering on supported devices. Euclid provides extensions for creating an `SCNGeometry`

from a `Mesh`

, as well as converting Euclid `Vector`

and `Rotation`

types to `SCNVector`

and `SCNQuaternion`

respectively.

The SceneKit integration makes it easy to display Euclid geometry on-screen, and to integrate with ARKit, etc. You can also use SceneKit to export Euclid-generated `Mesh`

es in standard 3D model formats such as DAE or OBJ.

# Materials

Interesting geometry is all well and good, but to really bring a shape to life it needs colors and textures.

Every `Polygon`

has a `material`

property that can be used to apply any kind of material you like on a per-polygon basis.

All primitives and builder methods accept a `material`

parameter which will apply that material to every polygon in the resultant `Mesh`

. When you later combine meshes using CSG operations, the original materials from the `Mesh`

es that contributed to each part of the resultant shape will be preserved.

Since Euclid knows nothing about the `material`

type, it can't do anything with it except pass it around. To use it with SceneKit you need to convert the Euclid material to an `SCNMaterial`

. This can be done using the optional closure argument for the Euclid's `SCNGeometry`

constructor, which receives the Euclid material as an input and returns an `SCNMaterial`

.

## Color

Euclid currently has no built-in concept of color, and no support for setting colors on a per-vertex basis, but you can apply colors to a `Mesh`

or `Polygon`

using the material property.

The material property is of type `AnyHashable`

which basically means it can be anything you want. Any `NSObject`

-derived class conforms to `AnyHashable`

, so a simple option is to set the `material`

to be a `UIColor`

or `NSColor`

.

This approach is demonstrated in the Example app included in the project.

## Textures

Euclid automatically adds 2D texture coordinates to the vertices of `Mesh`

es created using primitives or builder methods. There is limited control over how those coordinates are specified at the moment, but they allow for simple spherical and cylindrical texture wrapping.

To apply a texture image to a `Mesh`

, just store a `UIImage`

or `NSImage`

as the material property and then convert it to an `SCNMaterial`

using the same approach used for colors in the Example app.

If you want to do something more complex, such as applying both a color *and* texture to the same `Mesh`

, or maybe including a normal map or some other material properties, you could create a custom material type to store all the properties you care about, or even assign an `SCNMaterial`

directly as the material for your Euclid geometry.

# Credits

The Euclid framework is primarily the work of Nick Lockwood.

Special thanks go to Evan Wallace, whose JavaScript CSG library provided the inspiration for Euclid in the first place, along with the BSP algorithm used for Euclid's CSG operations.

Thanks also go to Patrick Goley, who first suggested "Euclid" for the library name.