# Involutes of Pseudo-Null Curves in Lorentz–Minkowski 3-Space

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Proposition**

**1.**

**Proof.**

**Corollary**

**1.**

**Proof.**

## 3. Involutes of a Pseudo-Null Curve

**Theorem**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

**Remark**

**1.**

## 4. Reconstruction by the Involutes

Question.For a given null straight line, is it possible to find a pseudo-null curve whose involute is the initial straight line?

**Theorem**

**2.**

**Proof.**

**Corollary**

**2.**

**Proof.**

**Corollary**

**3.**

**Proof.**

**Proposition**

**3.**

**Proof.**

**Remark**

**2.**

**Theorem**

**3.**

**Proof.**

**Example**

**4.**

**Example**

**5.**

**Example**

**6.**

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Examples 1 (

**left**), 2 (

**middle**) and 3 (

**right**). Example 1: The tangent developables as part of a lightlike plane. The tangent developable (8) with a base curve (red) and involutes (9) (green) for $a=-2,0,2$. Example 2: The developable tangent (11), with a base curve (red) and involutes (12) (green) for $a=-2,0,2$. Example 3: The developable tangent (14) with a base curve (red) and involutes (15) (green) for $a=-3.5,\phantom{\rule{0.166667em}{0ex}}-1,\phantom{\rule{0.166667em}{0ex}}1,\phantom{\rule{0.166667em}{0ex}}3.5$.

**Figure 2.**Example 4.1: the null straight line (green) is the involute of different pseudo-null curves (yellow, red, blue) for $a=1$ (and $b=3,\phantom{\rule{0.166667em}{0ex}}b=0,\phantom{\rule{0.166667em}{0ex}}b=-3$) (

**left**). The blue curve corresponds to the constant null frame, the red curve is (7) and the yellow curve corresponds to the not constant null frame. Example 4.2: the null straight line (green) is the involute of different pseudo-null curves (yellow, red, blue) for $a=0,\phantom{\rule{0.166667em}{0ex}}\tau =0.5$ and $b=2,\phantom{\rule{0.166667em}{0ex}}b=5,\phantom{\rule{0.166667em}{0ex}}b=6$ (

**right**). The yellow and red corresponds to the null frame (24) and the blue curve corresponds to the null frame (25).

**Figure 3.**Example 6: The null straight line (green) given by (15) and $a=-1$ is the involute of different pseudo-null curves. The yellow, red and blue curve correspond to the constant null frame with $b=6,\phantom{\rule{0.166667em}{0ex}}b=2,\phantom{\rule{0.166667em}{0ex}}b=9,$ respectively. The red curve is the curve (13) (

**Left**). (

**Right**): The null straight line (green) given by (15) and $a=2$ is the involute of the yellow and blue curve that correspond to the not constant null frame with $m=0,\phantom{\rule{0.166667em}{0ex}}{\kappa}_{3}=0.5,\phantom{\rule{0.166667em}{0ex}}b=10$ and $b=6$, as well as of the red curve given by $m=0,\phantom{\rule{0.166667em}{0ex}}{\kappa}_{3}=0,\phantom{\rule{0.166667em}{0ex}}b=2$ (that is the curve (13)).

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**MDPI and ACS Style**

López, R.; Milin Šipuš, Ž.; Primorac Gajčić, L.; Protrka, I.
Involutes of Pseudo-Null Curves in Lorentz–Minkowski 3-Space. *Mathematics* **2021**, *9*, 1256.
https://doi.org/10.3390/math9111256

**AMA Style**

López R, Milin Šipuš Ž, Primorac Gajčić L, Protrka I.
Involutes of Pseudo-Null Curves in Lorentz–Minkowski 3-Space. *Mathematics*. 2021; 9(11):1256.
https://doi.org/10.3390/math9111256

**Chicago/Turabian Style**

López, Rafael, Željka Milin Šipuš, Ljiljana Primorac Gajčić, and Ivana Protrka.
2021. "Involutes of Pseudo-Null Curves in Lorentz–Minkowski 3-Space" *Mathematics* 9, no. 11: 1256.
https://doi.org/10.3390/math9111256