Geometry in affiner

In affiner: A Finer Way to Render 3D Illustrated Objects in 'grid' Using Affine Transformations

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

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Table of Contents

• Angle objects
• Trigonometry
• 2D Coordinates
• 3D Coordinates
• Orthographic/Axonometric and Oblique Projections

Angle objects

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

• Supports the following angular 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.

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

Trigonometry

{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.

library("affiner")
sin(2 * pi)
sine(degrees(360))
arctangent(x = 0, y = 1)

2D Coordinates

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.

# 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()

3D Coordinates

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).

# 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)

Orthographic/Axonometric and Oblique Projections

{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") 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

affiner documentation built on April 4, 2025, 4:42 a.m.