
---
title: "Geometry in affiner"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Geometry in affiner}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

### Table of Contents

* [Angle objects](#angles)
* [Trigonometry](#trigonometry)
* [2D Coordinates](#2d)
* [3D Coordinates](#3d)
* [Orthographic/Axonometric and Oblique Projections](#projection)

## <a name="angles">Angle objects</a>

In `{affiner}` angles are represented by the `angle()` class:

* Supports the following [angular units](https://en.wikipedia.org/wiki/Angle#Units) (note we ignore any punctuation and space characters as well as any trailing s's e.g. "half turns" will be treated as equivalent to "halfturn"):

  + "deg" or "degree"
  + "half-revolution", "half-turn", or "pi-radian"
  + "gon", "grad", "grade", or "gradian"
  + "rad" or "radian"
  + "rev", "revolution", "tr", or "turn"

* `degrees()`, `gradians()`, `pi_radians()`, `radians()`, `turns()` are convenience wrappers around `as_angle.()` that specifies the angular unit. 

* One can use the `affiner_angular_unit` global option to set the default angular unit used by this package from "degrees" to "gradians", (multiples of) "pi-radians", "radians", or "turns".

* Use `is_congruent()` to check if two angles are congruent modulo full turns.

```{r angles}
library("affiner")
as_angle(90, "degrees") + turns(1)
is_congruent(degrees(180), radians(pi))
as.numeric(turns(1/3), "radians")
```

## <a name="trigonometry">Trigonometry</a>

`{affiner}` provides several `angle()` class aware trigonometric functions:

* `sine()`, `cosine()`, `tangent()`, `secant()`, `cosecant()`, `cotangent()`,
* `arcsine()`, `arccosine()`, `arctangent()`, `arcsecant()`, `arccosecant()`, and `arccotangent()`.
    `arcsine()` and `arccosine()` also feature a `tolerance` value so that values that exceed the `1` / `-1` cutoffs by a small tolerance are rounded to those values.
* One can also use the base S3 methods `sin()`, `cos()`, and `tan()` on `angle()` objects.

```{r trig}
library("affiner")
sin(2 * pi)
sine(degrees(360))
arctangent(x = 0, y = 1)
```

## <a name="2d">2D Coordinates</a>

In `{affiner}` 2D Coordinates are represented by a `Coord2D` R6 class:

* Create `Coord2D` objects with `as_coord2d()`
* `Coord2D` R6 objects supports several affine transformation methods that can be chained:

  + `permute()`
  + `project()`
  + `reflect()`
  + `rotate()`
  + `scale()`
  + `shear()`
  + `translate()`
  + `transform()`
  + R6 method chained affine transformation matrices are auto-multiplied
    so you don't need to manually multiply them for efficiency reasons.
  + `{affiner}` affine transformations are post-multiplied so affine transformations
    can be applied in an intuitive order.
  +  `abs()` computes Euclidean norm and `distance2d()` computes Euclidean distances
  + `convex_hull2d()` computes the convex hull.
  * `range()` computes the axis-aligned bounding box ranges.

```{r 2d}
# Cartesian coordinates
library("affiner")
p <- as_coord2d(x = 1:10, y = 1:10)
print(p)

p2 <- p$
    clone()$
    scale(x = 0.5)$
    rotate(degrees(90))$
    reflect(as_line2d("y-axis"))$
    translate(as_coord2d(x = 0.5, y = 0.5))$
    print()

# Polar coordinates
theta <- degrees(seq(0, 300, by = 60))
radius <- 1
p <- as_coord2d(theta, radius = radius)
is_congruent(as_angle(p), theta) |> all()
is_congruent(abs(p), radius) |> all()
```

## <a name="3d">3D Coordinates</a>

In `{affiner}` 3D Coordinates are represented by a `Coord3D` R6 class:

* Create `Coord3D` objects with `as_coord3d()`
* `Coord3D` R6 objects supports several affine transformation methods that can be chained:

  + `permute()`
  + `project()`
  + `reflect()`
  + `rotate()`
  + `scale()`
  + `shear()`
  + `translate()`
  + `transform()`
  + R6 method chained affine transformation matrices are auto-multiplied
    so you don't need to manually pre-multiply them for efficiency reasons.
  + `{affiner}` affine transformations are post-multiplied so affine transformations
    can be applied in an intuitive order.
  +  `abs()` computes Euclidean norm and `distance3d()` computes Euclidean distances
  * `range()` computes the axis-aligned bounding box ranges.
  * `cross_product3d()` computes cross products (`*` computes inner products).

```{r 3d}
# Cartesian coordinates
library("affiner")
p <- as_coord3d(x = 1:10, y = 1:10, z = 1:10)
print(p)

p2 <- p$
    clone()$
    scale(z = 0.5)$
    rotate(axis = as_coord3d("z-axis"), theta = degrees(90))$
    reflect(as_plane3d("yz-plane"))$
    shear(xy_shear = 0.5)$
    translate(as_coord3d(x = 0.5, y = 0.5, z = 0.5))$
    print()

# Spherical coordinates
inclination <- as_angle(p, type = "inclination")
azimuth <- as_angle(p, type = "azimuth")
radius <- abs(p)
ps <- as_coord3d(azimuth, radius = radius, inclination = inclination)
all.equal(p, ps)

# Cylindrical coordinates
radius <- as_coord2d(p, plane = "xy-plane") |> abs()
pc <- as_coord3d(azimuth, radius = radius, z = p$z)
all.equal(p, pc)
```

## <a name="projection">Orthographic/Axonometric and Oblique Projections</a>

`{affiner}` can project `Coord3D` objects to `Coord2D` objects using orthographic/axonometric and oblique projections:

* For a multiview/primary orthographic projection onto the xy-plane use `as_coord2d(x)`
* For a multiview/primary orthographic projection onto the xz-plane use `as_coord2d(x, permutation = "xzy")`
* For a "cabinet" oblique projection onto the xy-plane use `as_coord2d(x, scale = 0.5)`
* For a "cabinet" oblique projection onto the xz-plane use `as_coord2d(x, permutation = "xzy", scale = 0.5)`
* For other oblique projections manipulate the `scale` parameter (usually from 0.5 to 1.0) and the `alpha` angle parameter
  (usually from 30° to 45°).
* For one "isometric" axonometric projection one can use

  ```
  x$
    clone()$
    translate(-mean(x)$
    rotate("z-axis", degrees(45))$
    rotate("x-axis", degrees(-90 + 35.264)) |>
    as_coord2d()
  ```

* Other axonometric projections can be achieved with the right 3D rotations
* See [`vignette("affiner", package = "affiner")`](affiner.html) for some visual examples
* Recall one can use "scale" affine transformation to flip signs of x/y/z axes and "permute" affine transformation to switch order of x/y/z coordinates
