# Find the equation of a curve between two points on a cone respectively on a cylinder and calculate a new point on this curve

I have a curve between two points $$P_1 (x_1, y_1,z_1)$$ and $$P_2 (x_2, y_2, z_2)$$ which both lie either on the surface of a cylinder (case $$A$$) or on the surface of a cone (case $$B$$). The curve is the shortest possible curve between $$P_1$$ and $$P_2$$.

It is given:

A. cylinder

• center axis is identical with the $$x$$-axis
• radius is $$R$$

B. cone

• center axis is identical with the $$x$$-axis
• vertex is in $$x=0, y=0, z=0$$
• radius at $$x=H (H>0)$$ is $$R$$

Furthermore a distance $$D$$ is given. $$D$$ is smaller than the length of the curve between $$P_1$$ and $$P_2$$.

I need to calculate the $$(x,y,z)$$ coordinates of a new point $$V$$ on the curve between $$P_1$$ and $$P_2$$ at the distance $$D$$ from $$P_1$$ i.e. the curve length between $$P_1$$ and $$V$$ should be $$D$$.

The curve between $$P_1$$ and $$P_2$$ does not pass over the end caps or through the cone vertex.

How do I calculate $$(x,y,z)$$ of the new point $$V$$ in case $$A$$ and case $$B$$?

• @JeanMarie Thanks for the outline. Could you please in more detail how to perform these steps? Thanks. Jun 16, 2023 at 22:37
• Sorry, I forgot to specify that the curve between P1 and P2 should be the shortest possible curve between the two points. I have completed the question accordingly. Jun 16, 2023 at 22:40
• If it is a shortest path, see this answer Jun 17, 2023 at 5:28

HINT:

It is very convenient to consider the development of a cone. Helices on 3d cone surface turn out to be straight lines on the sector of circle when laid out flat. I prefer to use uncapitalized symbol $$r$$ in place of $$R$$;

I describe briefly in the hints. Shall be glad to explain if required in more detail. It would be useful to derive geodesic coordinates of the helix using Claiaut's Law.

Mathcurve Ref StrLine & Helix

$$\frac{r}{\rho}=\sin \alpha = \frac{\theta_{subtended~at ~cone ~apex}}{\theta_{rotation~ in ~cone~ base}};\tan \alpha =\frac{r_{max}}{H};$$

$$(r,\rho)$$ are radii, slant generator in 2d, 3d respectively on cylinder and cone. $$(P1,V, P2)$$ lie on the cone development straight line geodesic , as well as on the 3d helix. Same label is used. $$V$$ is a variable point on the straight line and helix. The sector can be rolled back to cone by bending along cone generators by the above relation, getting $$(x,y,z)$$ from $$(\rho, \theta_{subtended ~at ~cone ~apex})$$. Such rolling /development retains the shortest path character between them.

The cylinder is a special case of the cone when $$\alpha=0$$.

• Thank you for the hint. However I do not see how to calculate (x,y,z) of the new point V in detail. Therefore, I would like to accept your offer and ask for more detailed explanations, i.e the actual calculation of the (x,y,z) of V. Thank you! Jun 17, 2023 at 19:20
• Did you see the Mathcurve Link (geodesics)? Next can you change from given polar to required cartesian coordinates? In this site actually we like to see more work done by OPs towards solution. Jun 17, 2023 at 19:49
• Yes, I did see the Mathcurve Link. Thanks. How do I change from P (x,y,z) to polar coordinates and back? I am not an expert. I am trying to learn. Sorry. Jun 17, 2023 at 20:11