Arithmetic: Arithmetic operators on SO(3)

In rotations: Working with Rotation Data

Arithmetic operators on SO(3)

Description

These binary operators perform arithmetic on rotations in quaternion or rotation matrix form (or objects which can be coerced into them).

Usage

## S3 method for class 'SO3'
x + y

## S3 method for class 'SO3'
x - y = NULL

## S3 method for class 'Q4'
x + y

## S3 method for class 'Q4'
x - y = NULL

Arguments

x	first argument
y	second argument (optional for subtraction)

Details

The rotation group SO(3) is a multiplicative group so “adding" rotations R_1 and R_2 results in R_1+R_2=R_2R_1. Similarly, the difference between rotations R_1 and R_2 is R_1-R_2=R_2^\top R_1. With this definition it is clear that R_1+R_2-R_2=R_2^\top R_2R_1=R_1. If only one rotation is provided to subtraction then the inverse (transpose) it returned, e.g. -R_2=R_2^\top.

Value

+	the result of rotating the identity frame through x then y
-	the difference of the rotations, or the inverse rotation of only one argument is provided

Examples

U <- c(1, 0, 0)          #Rotate about the x-axis
R1 <- as.SO3(U, pi/8)    #Rotate pi/8 radians about the x-axis
R2 <- R1 + R1            #Rotate pi/8 radians about the x-axis twice
mis.axis(R2)             #x-axis: (1,0,0)
mis.angle(R2)            #pi/8 + pi/8 = pi/4

R3 <- R1 - R1            #Rotate pi/8 radians about x-axis then back again
R3                       #Identity matrix

R4 <- -R1                #Rotate in the opposite direction through pi/8
R5 <- as.SO3(U, -pi/8)   #Equivalent to R4

M1 <- matrix(R1, 3, 3)   #If element-wise addition is requred,
M2 <- matrix(R2, 3, 3)   #translate them to matrices then treat as usual
M3 <- M1 + M2

M1 %*% M1                #Equivalent to R2
t(M1) %*% M1             #Equivalent to R3
t(M1)                    #Equivalent to R4 and R5

#The same can be done with quaternions: the identity rotation is (1, 0, 0, 0)
#and the inverse rotation of Q=(a, b, c, d) is -Q=(a, -b, -c, -d)

Q1 <- as.Q4(R1)
Q2 <- Q1 + Q1
mis.axis(Q2)
mis.angle(Q2)

Q1 - Q1                  #id.Q4 = (1, 0, 0, 0)

rotations documentation built on May 29, 2024, 6:02 a.m.