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Question: Calculating Slip Between Two Curves at Their Contact Point in Space

I want to calculate the slip between two curves at their contact point in space as they move and/or stretch. The curves are touching at only one point, and this contact point might change or might not change. Of course, slip can still occur at the contact point even if the contact point did not change in space.

For example, let's assume for simplicity that these two curves are lines.

Line One:

$ \mathbf{L1}(t) = \left[ A_x t + B_x, A_y t + B_y, A_z t + B_z \right]$

Line Two:

$ \mathbf{L2}(t) = \left[ C_x t + D_x, C_y t + D_y, C_z t + D_z \right]$

After movement:

Line One:

$ \mathbf{L1'}(t) = \left[ A_x' t + B_x', A_y' t + B_y', A_z' t + B_z' \right]$

Line Two:

$ \mathbf{L2'}(t) = \left[ C_x' t + D_x', C_y' t + D_y', C_z' t + D_z' \right]$

Their contact points before and after movement:

$ \mathbf{CP} = \left[ \text{CP}_x, \text{CP}_y, \text{CP}_z \right]$

$ \mathbf{CP'} = \left[ \text{CP}_x', \text{CP}_y', \text{CP}_z' \right]$

Given all the information before and after the movement, how do I calculate the relative slip between these two curves at their contact point?

Clarifications and Constraints:

  1. The contact point may or may not change spatially.
  2. Slip is considered to occur at the contact point even if the contact point remains the same in space.
  3. I need a method to quantify this slip using the provided line equations before and after movement.

Any insights or detailed explanations would be greatly appreciated!

edit1: t is a spatial parameter that varies along the curves, for example, ranging from 0 to 100, where it represents the position of points on the curves over the specified interval

edit2: to understand what I mean by slip: As an analogy, consider a knife cutting a cylinder. The cylinder lies on the X-axis, and the knife moves in various directions. If the knife is parallel to the Y-axis and moves upward by one unit, it travels one unit on the cylinder's surface. However, if the knife is at a 45-degree angle to the X-axis and moves upward by one unit, the distance it travels on the surface of the cylinder might be different. Understanding this distance is crucial to calculating the wear, as it directly relates to how much material is being cut or displaced.

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  • $\begingroup$ t is a spatial parameter that varies along the curves, for example, ranging from 0 to 100, where it represents the position of points on the curves over the specified interval. $\endgroup$
    – Oday Allan
    Commented Jun 11 at 11:35
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    $\begingroup$ Do you have a mathematical definition of slip? Obviously not in terms of these quantities, but in terms of concepts that we can understand unambiguously. $\endgroup$ Commented Jun 11 at 12:39
  • $\begingroup$ I want to use this slip to calculate surface degradation, such as using the Archard wear equation. Essentially, I need to determine how much the contact point has moved relative to each curve. I'm not exactly sure how to define this in robust mathematical terms. However, if I can track the path of the contact point relative to one curve and integrate this path, it would essentially give me the measure of slip I'm talking about. $\endgroup$
    – Oday Allan
    Commented Jun 11 at 12:51

1 Answer 1

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Here's one idea. Let's define the $t$ spatial parameter values for the contact points on the two curves before and after movement as follows, \begin{gather} CP = L1(t_1) = L2(t_2) \\ CP' = L1'(t'_1) = L2'(t'_2) \end{gather}

Then you could define relative slip as the ratio of the distances travelled by the contact points on the two curves due to 1) curve movement in space, and 2) change in contact point $$\frac{|L1'(t_1) - L1(t_1)| + |L1'(t'_1) - L1'(t_1)|}{|L2'(t_2) - L2(t_2)| + |L2'(t'_2) - L2'(t_2)|}$$

where $|P - Q|$ is the Euclidean distance between two points, calculated as $$|P - Q| \equiv \sqrt{(P_x - Q_x)^2 + (P_y - Q_y)^2 +(P_z - Q_z)^2}$$

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  • $\begingroup$ It does not work for a very simple case of having line 1 lie on the X axis and line 2 lies on the Y axis. now if line 2 moves (N) units along the Y axis, the slip as per your solution will always amount to zero, as the contact point did not move in space, nor did curve 1 move in space, so L1 = L1' and t1 = t1'. $\endgroup$
    – Oday Allan
    Commented Jun 12 at 8:53

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