# Geometric Properties of Planar and Spherical Interception Curves

## Abstract

**:**

## 1. Introduction

## 2. Planar Curve

**Question 1.**Suppose that two points, $P(x,y)$ and Q, initially at $O(0,0)$ and $A(1,0)$, respectively, move with constant and equal velocities so that Q is on the line $x=1$ and P is on the ray $OQ$. What curve is defined by point P?

- Use substitution ${r}^{\prime}=rcotu$ to obtain $r{cos}^{2}\theta =\pm asinu$. By differentiating and excluding r and ${r}^{\prime}$, we obtain ${u}^{\prime}+2tanutan\theta =1$ ([10], Section 1.460);
- Use substitutions $\eta \left(\xi \right)=tanu$, $\xi =tan\theta $ to obtain Abel’s equation $({\xi}^{2}+1){\eta}^{\prime}=({\eta}^{2}+1)(1-2\xi \eta )$ ([10], Section 1.81);
- Use substitution ${\xi}^{4}\eta \left(\xi \right)=({\xi}^{2}+1)z+{\xi}^{3}$ to obtain one more Abel’s equation ${\xi}^{7}{z}^{\prime}+2({\xi}^{2}+1){z}^{3}+5{\xi}^{3}{z}^{2}=0$ ([10], Section 1.151);
- Use substitution $v=\frac{1}{z}$ to obtain ${\xi}^{7}v{v}^{\prime}=2({\xi}^{2}+1)+5{\xi}^{3}v$ ([10], Section 1.185);
- Use substitution $\xi w={\xi}^{3}v+1$ to obtain linear equation $\frac{d\xi}{dw}-\frac{\xi w}{2({w}^{2}+1)}+\frac{1}{2({w}^{2}+1)}=0$, which can be solved by quadratures ([10], Section 1.185).

**Answer to Question 1.**Now, let us try using cartesian coordinates to find the solution of Equation (2) as a parametric curve. First, note that in the cartesian coordinates, Equation (1) can be written as

**Theorem 1.**

- 1
- $\left|UP\right|=\left|OU\right|+\left|TQ\right|$,
- 2
- $(1-x)\xb7\left|UP\right|=\left|TQ\right|$,
- 3
- $x\xb7\left|PT\right|=\left|TQ\right|$,
- 4
- $sin\angle QPT=\left|OP\right|\xb7{sin}^{2}\angle TQP$,

**Proof.**

**Theorem 2.**

**Proof.**

**Theorem 3.**

**Proof.**

## 3. Spherical Curve

**Question 2.**Suppose that two points P and Q, initially at $B(0,0,1)$ and $A(1,0,0)$, respectively, move with constant and equal velocity so that Q is on the great circle $z=0$, ${x}^{2}+{y}^{2}=1$ of sphere ${x}^{2}+{y}^{2}+{z}^{2}=1$ with center $O(0,0,0)$, and P is on the great circle through B and Q of the sphere. What curve is defined by point P? (see Figure 3).

**Answer.**We use spherical coordinates to describe the resulting interception curve. We denote $\angle AOQ=\theta $ and $\angle POB=\varphi $. Since $\rho =\left|OP\right|=1$, for the coordinates of point $P(x,y,z)$, we can write $x=cos\theta sin\varphi $, $y=sin\theta sin\varphi $, and $z=cos\varphi $, where we assume that $\varphi =\varphi \left(\theta \right)$ is a function of $\theta $. Therefore, ${x}_{\theta}^{\prime}=-sin\theta sin\varphi +{\varphi}^{\prime}cos\theta cos\varphi $, ${y}_{\theta}^{\prime}=cos\theta sin\varphi +{\varphi}^{\prime}sin\theta cos\varphi $, and ${z}_{\theta}^{\prime}=-{\varphi}^{\prime}sin\varphi $. Since the points P and Q travel equal distances, by using the well-known formula from vector calculus for the length of a parametrically defined curve, we can write

**Theorem 4.**

**Proof.**

**Theorem 5.**

- 1
- the sum of the lengths of the arcs $PT$ and $TQ$ is not dependent on θ, and $\widehat{PT}+\widehat{TQ}=\frac{\pi}{2}$,
- 2
- as θ increases, $\widehat{TQ}$ increases, $\widehat{PT}$ decreases, and both approach to $\frac{\pi}{4}$, as $\theta \to \infty $.
- 3
- spherical angle $\angle QPT$ is equal to $\widehat{BP}$,
- 4
- spherical angle $\angle BPT$ is equal to $\widehat{PQ}+\frac{\pi}{2}$.

**Proof.**

**Theorem 6.**

## 4. Alternative Methods for Spherical and Planar Curves

**Lemma 1.**

**Proof.**

**Lemma 2.**

**Lemma 3.**

- 1
- $\left|OU\right|+|T{Q}_{2}|=|U{P}_{1}|$ and $\left|OU\right|+|T{Q}_{1}|=|U{P}_{2}|$,
- 2
- the radii of circles through points $O,{P}_{1},{P}_{2}$ and $O,{Q}_{1},{Q}_{2}$ are equal,
- 3
- if the distance between the parallel lines is 1, and the distances from points ${P}_{1}$ and ${P}_{2}$ to line $OU$ are ${x}_{1}$ and ${x}_{2}$, respectively, then$$sin\angle {Q}_{1}{P}_{1}T\xb7sin\angle {Q}_{2}{P}_{2}T=|O{P}_{1}|\xb7|O{P}_{2}|\xb7{sin}^{2}\angle T{Q}_{1}{P}_{1}\xb7{sin}^{2}\angle T{Q}_{2}{P}_{2},$$$${x}_{1}{x}_{2}\xb7|{P}_{1}T|\xb7|{P}_{2}T|=|T{Q}_{1}|\xb7|T{Q}_{2}|,$$$$(1-{x}_{1})(1-{x}_{2})\xb7|U{P}_{1}|\xb7|U{P}_{2}|=|T{Q}_{1}|\xb7|T{Q}_{2}|.$$

## 5. Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Table A1.**Corresponding theorems in planar and spherical cases, and lemmata in Section 4 that can be used to prove these results.

Plane Curve | Spherical Curve | (A)symmetry | Plane Geometry | Spherical Geometry |
---|---|---|---|---|

Theorem 1 | Theorem 5 | Symmetry | Lemma 3, 1. & 3. | Lemma 1 |

Theorem 2 | Theorem 6 | Asymmetry | Lemma 3, part 2. | Lemma 2 |

Theorem 3 | Theorem 4 | Asymmetry | - | - |

**Table A2.**Corresponding curves on the sphere, on the cylinder and on the plane in Figure 4.

Sphere ${\mathit{x}}^{2}+{\mathit{y}}^{2}+{\mathit{z}}^{2}=1$ | Cylinder ${\mathit{x}}^{2}+{\mathit{y}}^{2}=1$ Mercator Projection | Plane $\mathit{z}=0$ Stereographic Projection |
---|---|---|

$\varphi =2{tan}^{-1}{e}^{\theta}-\frac{\pi}{2}$ Interception curve (blue) | $y=lncoth\frac{x}{2}$ (green) | $r=coth\frac{\theta}{2}$ (black) |

$\left(\right)$ Spherical spiral (red) | $y=x$ Helix (orange) | $r={e}^{\theta}$ Logarithmic spiral (purple) |

## Appendix B

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**Figure 1.**Interception curve (blue) on a Plane. Its tangent line (black), the line containing its position vector (black), and the line $x=1$ (blue) are also shown.

**Figure 2.**Comparison of interception (left, red) and pursuit (right, red) curves. Created using parametrization (5) and Bouguer’s formula. The tangent lines (green), the lines containing the position vectors (thin blue) of the curves, and lines $x=1$ (thick blue) are also shown.

**Figure 3.**Interception curve (light blue) on a Unit Sphere. Its tangent great circle (dark blue), meridian (red) and equator (black) great circles are also shown.

**Figure 4.**Mercator and stereographic projections of the spherical interception curve (blue) and spherical spiral (red). The other colors are explained in the text and in Appendix A, Table A1.

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**MDPI and ACS Style**

Aliyev, Y.N.
Geometric Properties of Planar and Spherical Interception Curves. *Axioms* **2023**, *12*, 704.
https://doi.org/10.3390/axioms12070704

**AMA Style**

Aliyev YN.
Geometric Properties of Planar and Spherical Interception Curves. *Axioms*. 2023; 12(7):704.
https://doi.org/10.3390/axioms12070704

**Chicago/Turabian Style**

Aliyev, Yagub N.
2023. "Geometric Properties of Planar and Spherical Interception Curves" *Axioms* 12, no. 7: 704.
https://doi.org/10.3390/axioms12070704