Solving Planer Curves Locus with Matrix M_{2f}

[bugatti79]

Hi Folks,

I am puzzled how entries M_13 and M_23 were obtained in the following matrix $$M_{2f}$$ as in attached picture.

Coordinate systems S1 is attached to bottom left circle, S2 to top left circle and Sf is rigidly attached to the frame.

If we assume $$\phi_{2}=0$$ we can see that entry 31 becomes 0 and entry becomes -C. How are these obtained?
I would have thought based on schematic that the y2=y1+C but it appears from the matrix that y2=y1-C...

Any thoughts?

 

[jvicens]

Hello,

Thank you for your post. Based on the information provided, it seems that the matrix M_{2f} represents a transformation between two coordinate systems, S1 and S2, which are attached to two circles and a third coordinate system Sf attached to the frame. The entries M_13 and M_23 in the matrix represent the translation components in the x and y directions, respectively.

To obtain these values, we need to consider the relative positions and orientations of the three coordinate systems. Since Sf is rigidly attached to the frame, its origin will always remain fixed in the frame's reference frame. However, the other two coordinate systems, S1 and S2, can move and rotate with respect to the frame.

In this case, the entry M_13 represents the distance between the origins of S1 and Sf in the x direction. Similarly, M_23 represents the distance between the origins of S2 and Sf in the y direction. These values can be calculated using basic geometry and trigonometry, taking into account the angles and distances between the coordinate systems.

Additionally, it seems that the value of \phi_2=0 means that the two circles are aligned in the same orientation, which simplifies the calculations. However, it is important to note that the values of y2 and y1 do not necessarily have to be equal, as they represent the distances between the origins of S2 and S1, respectively, and the origin of Sf. So, it is possible for y2 to be equal to y1+C or y1-C, depending on the specific positions and orientations of the coordinate systems.

I hope this helps to clarify the situation. Please let me know if you have any further questions or if I can assist you in any other way.