Help with 3d trig and moving object

[free-node-5]

first off, sorry if this isn't the correct section for this question
maybe if it's not, someone could move it to a better place
It's not for any class but someone told me that it was similar to mechanics

I have an object in 3d space, say a cone for example, with the tip pointing up (positive Z direction)
The plane on which it rests is the x,y plane

It can have rotations of the form (x,y,z) in radians

To begin, a specific direction is chosen as a Z angle, and the object is to move in the path of an arc through a 2d path, but in the direction of the Z angle

I managed to come up with and test some equations for moving the object along it's path, but I need some equations to make it point (rotation) in the direction that it is moving on the arc

(up at the beginning, sideways in the middle, and down at the end)

The movement is controlled by using an independent variable 't' as the radians traveled on the arc

Here are the equations I'm using for the motion:
xComponent = Cos(theta)
yComponent = Sin(theta)

x = radius * xComponent * Cos( t )
y = radius * yComponent * llCos( t )
z = radius * Sin( t )


Can anyone help me figure out the equations for the rotations?
Any help would be greatly appreciated
Thanks

 

[eljose79]

Thank you for reaching out with your question. From what I understand, you are trying to move an object in 3D space along an arc while also controlling its rotation along the arc. This can be a challenging problem, but I will do my best to provide some guidance.

Firstly, it is difficult to provide specific equations without knowing the exact details of your setup and the constraints you have. However, I can offer some general advice on how to approach the problem.

Since you are already using an independent variable 't' to control the motion along the arc, you could also use this variable to control the rotation. One way to do this is by defining a rotation matrix that takes into account the current value of 't' and rotates the object accordingly.

For example, if you want the object to rotate along the arc in the direction of the Z angle, you could define a rotation matrix as follows:

R = [cos(t) -sin(t) 0;
sin(t) cos(t) 0;
0 0 1]

This matrix represents a rotation around the Z axis by an angle of 't'. You can then use this matrix to transform the object's orientation at each time step.

Another approach could be to define a function that takes in the current value of 't' and outputs the desired rotation for the object. This function could be based on the specific path and motion you want the object to follow.

I would also recommend looking into the concept of quaternions, which are commonly used in 3D graphics and animation to represent rotations. They can provide a more intuitive way to control the orientation of an object along a path.

I hope this helps and provides some direction for you. If you have any further questions, please don't hesitate to ask. Good luck with your project!

 

[charng]

I would suggest using vector calculus and kinematics equations to solve for the rotations of the object. This involves considering the velocity and acceleration vectors of the object as it moves along the arc.

One approach would be to use the tangent vector of the arc at each point to determine the direction the object should be facing. This can be calculated by taking the derivative of the position vector with respect to time. Then, use the cross product of the tangent vector and the initial orientation of the object to determine the axis of rotation. Finally, use the angle between the initial orientation and the tangent vector to determine the amount of rotation needed.

Another approach would be to use quaternions, which are a mathematical tool commonly used in 3D rotations. Quaternions can represent rotations in 3D space and can be used to calculate the orientation of the object at any point along the arc.

I would also recommend consulting with a mathematician or a physics expert for further assistance with the equations. Good luck with your project!