Understanding 3D Rotation Transformations

In summary, the conversation discusses confusion about rotation transformations and how trigonometry works in 3D. The main question is how the conventional rules for 2D rotation translate to other perspectives of the 3 cartesian axes. The conversation also mentions the use of matrix transformations to control the orientation of a 3D shape.
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student6587
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Hello,

First time posting to Physics Forums.

I have been thinking about rotation transformations and am a bit confused on how trig works in 3D.

In 2D, convention says the positive x-axis points to the right, the positive y-axis points upward, and positive angles are measured from the positive x-axis in a counter-clockwise fashion. Proper insertion of a third dimension has the positive z-axis pointing toward the viewer.

How do these rules translate to other perspectives of the 3 cartesian axes? For example, if the positive x-axis points to the left, the positive z axis points up, and the positive y-axis points toward the viewer. Is positive rotation still counter-clockwise? What axis is this angle measured from?

I suspect that the convention is arbitrary but there must be some criteria for consistency. A little bit of context: ultimately, I want to use this knowledge to construct matrix transformations to control the orientation of a simple 3D shape. When I try to work these out by hand, I keep getting the signs wrong.

Thanks!
 
Mathematics news on Phys.org

What are 3D rotation transformations?

3D rotation transformations are mathematical operations that are used to manipulate the position and orientation of objects in a 3D space. They involve rotating an object around one or more axes, and can be used to create a variety of effects in computer graphics and animation.

What is the difference between Euler angles and quaternions in 3D rotation transformations?

Both Euler angles and quaternions are mathematical representations of 3D rotations. Euler angles use three separate angles to describe the rotation around each axis, while quaternions use a single complex number. Quaternions are generally preferred for 3D rotations as they avoid the problem of gimbal lock and provide smoother interpolation between rotations.

How do 3D rotation matrices work?

3D rotation matrices are a mathematical tool used to perform 3D rotations. They are square matrices with special properties that allow them to represent rotations around a given axis. By multiplying a vector representing the position of an object by a rotation matrix, the position of the object can be transformed in 3D space.

What is the order of operations for 3D rotation transformations?

The order of operations for 3D rotation transformations is important and can affect the final result. Generally, rotations around different axes are performed in a specific order, such as ZYX or XYZ. This ensures that the rotations are applied in the correct order and produce the desired result.

How are 3D rotation transformations used in computer graphics?

3D rotation transformations are an essential tool in computer graphics, as they allow for the creation of complex 3D scenes and animations. They are used to manipulate the position and orientation of objects, create realistic movements, and create special effects. They are also used in video games, virtual reality, and other applications that require 3D graphics.

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