Z Rotation Cubing

4/15/2022 admin

Rotation Matrices

Calculates the new coordinates by rotation of points around the three principle axes (x,y,z). New coordinates by 3D rotation of points Calculator - High accuracy calculation Welcome, Guest.
I have a point (x,y,z) Suppose I make a 1 by 1 by 1 cube by making eight points at x.5, y.5, z.5 x +.5, y.5, z.5 x.5, y.5, z+.5 ect what would the formulas be for rotating this cube around the axises of the original point x,y,and z in any combination? (its position on the coordinate plane are totally irrelevant for the program i'm using.).

Z Rotation Cubing

Cube Rotations These are not actual turns, and is meant to instruct a rotation of whole the cube. Sometimes making a cube rotation allows executing much more convenient turns, like turning the R face, instead of the B face. There are 3 possible axes to rotate the cube on: The X, Y & Z axes. Learn 3x3 Rubik's cube in HINDI Easiest-Welcome back to another exciting video, in this video I showed you h. The x, y, and z rotations will be the same rotations as used in FCN. My method is based off of two very easy algorithms. They're both four moves, and they are the same as the 'sledgehammer' (R' F R F') and the 'hedgeslammer' (F R' F' R) on a 3x3x3 cube. Note that F' L F L' is the same as y' R' F R F', and L F' L' F is the same as y' F R' F' R.

Rotation matrices are used to rotate a vectorinto a new direction.

In transforming vectors in three-dimensional space, rotation matrices are often encountered. Rotation matrices are used in two senses: they can be used to rotate a vector into a new position or they can be used to rotate a coordinate basis (or coordinate system) into a new one. In this case, the vector is left alone but its components in the new basis will be different from those in the original basis. In Euclidean space, there are three basic rotations: one each around the x, y and z axes. Each rotation is specified by an angle of rotation. The rotation angle is defined to be positive for a rotation that is counterclockwise when viewed by an observer looking along the rotation axis towards the origin. Any arbitrary rotation can be composed of a combination of these three (Euler’s rotation theorem). For example, you can rotate a vector in any direction using a sequence of three rotations: $v^{'} = A v = R_{z} (γ) R_{y} (β) R_{x} (α) v$ .

The rotation matrices that rotate a vector around the x, y,and z-axes are given by:

Counterclockwise rotation around x-axis
Counterclockwise rotation around y-axis
Counterclockwise rotation around z-axis

The following three figures show what positive rotations looklike for each rotation axis:

For any rotation, there is an inverse rotation satisfying $A^{- 1} A = 1$ . For example, the inverse of the x-axis rotation matrix is obtained by changing the sign of the angle:

This example illustrates a basic property: the inverse rotation matrix is the transpose of the original. Rotation matrices satisfy A’A = 1, and consequently det(A) = 1. Under rotations, vector lengths are preserved as well as the angles between vectors.

We can think of rotations in another way. Consider the original set of basis vectors, $i, j, k$ , and rotate them all using the rotation matrix A. This produces a new set of basis vectors $i^{'}, j,^{'} k^{'}$ related to the original by:

Using the transpose, you can write the new basis vectors as a linear combinations of the old basis vectors:

Now any vector can be written as a linear combination of either set of basis vectors:

Using algebraic manipulation, you can derive the transformation of components for a fixed vector when the basis (or coordinate system) rotates. This transformation uses the transpose of the rotation matrix.

The next figure illustrates how a vector is transformed as the coordinate system rotates around the x-axis. The figure after shows how this transformation can be interpreted as a rotation of the vector in the opposite direction.

Projecting 3D on 2D

A point in a three-dimensional coordinate system can be represented by a (3times 1) matrix. When modelling three dimensions on a two-dimensional computer screen, you must project each point to 2D. After the projection, each point is represented by a (2times 1) matrix.

If we assume that the (z)-axes (or the third base vector) is pointing from the computer screen, then the screen is a projection on the (xy)-plane.

The projection matrix (mathbf{P}) must map the matrix (mathbf{v}=left( begin{array}{} abc end{array}right)) onto a matrix (mathbf{w}=left( begin{array}{} ab end{array}right)). This can be achieved by letting

[mathbf{P}=left( begin{array}{} 1 & 0 & 00 & 1 & 0 end{array}right)]

Then let (mathbf{w}=mathbf{P}mathbf{v}).

Matrix as a visual point

In GeoGebra, matrices are represented by lists. A (2times 1) matrix is not shown in the drawing pad.

If you want to show a matrix (mathbf{matrix1}=binom{2}{1}) as a point, you must use the matrix elements as coordinates for a new point. You retrieve an element of a matrix by using the command Element[<matrix>, <row>, <column>].

To show a (2times 1)-matrix (mathbf{w}) as a point (A), you write:

Exercise 1

Make three sliders to represent the matrix elements (v_x, v_y, v_z) of a matrix (mathbf{v}).
Create the matrix (mathbf{v}) by using the spreadsheet or by entering it as a list in the input bar: v={{v_x},{v_y},{v_z}}
Create the projection matrix (mathbf{P}).
Create the projected matrix (mathbf{w}=mathbf{P}mathbf{v}) in the input bar: w=P v
Create a point A, having the elements of (mathbf{w}) as coordinates.
Change the sliders. When changing the (v_z)-slider, nothing should happen with the point (A).

The rotations

Z Rotation Cubing

If the z-axes points from the screen, and if you rotate the base vectors by an angle (alpha) around the z-axes, you get the rotated vectors as in the picture below.

The third base vector, that is parallel to the z-axes, does not change.

The rotation matrix (mathbf{T_z}) for rotation around the z-axes is:

[mathbf{T_z}=left( begin{array}{ccc} cos alpha & -sin alpha & 0 sin alpha & cos alpha & 0 0 & 0 & 1 end{array} right) ]

In a similar way you can find the rotation matrices for rotation around the x- and the y-axes.

[mathbf{T_x}=left( begin{array}{ccc} 1 & 0 & 0 0 & cos alpha & -sin alpha 0 & sin alpha & cos alpha end{array} right) ]

[mathbf{T_y}=left( begin{array}{ccc} cos alpha & 0 & sin alpha 0 & 1 & 0 -sin alpha & 0 & cos alpha end{array} right) ]

Exercise 2

Enter three sliders representing the three rotation angles, one angle per axes/base vector.
Enter three rotation matrices, one matrix for each angle.
Instead of just projecting the matrix (mathbf{v}) to 2D, it must now be rotated before the projection. Change the definition of the matrix (mathbf{w}) from (mathbf{w}=mathbf{Pv}) to (mathbf{w}=mathbf{PT_xT_yT_zv})

The Vertices of the Cube

The eight vertices of the cube must be transformed in the same way as the matrix v. The vertices must be rotated, projected, and represented by visual points. Since the same operations must be applied eight times, it is easiest to use the spreadsheet.

Let the cube have its vertices in at the coordinates (1,1,1), (1,-1,1), (-1,1,1), (-1,-1,1), (1,1,-1), (1,-1,-1), (-1,1,-1) and (-1,-1,-1).

One way to do the operations is to enter the coordinates in the spreadsheet and then:

Make a matrix of the coordinates of one point, then make relative copies of that matrix.
Perform the transformations on one matrix, then make relative copies of the result.
Make a point of one transformed matrix, then make relative copies of the point.

The operations are done in column A in the picture below. The relative copies are made by dragging to the right.

Rotate the points to fill in the sides of the cube by using the Polygon tool.

It is of course possible to use other coordinates for the vertices of the cube.

Using the same spreadsheet, you can now rotate other 3D objects and project them on 2D.