On Geometry of Isophote Curves in Galilean space

TL;DR

This paper explores the geometry of isophote curves on surfaces in Galilean 3-space, analyzing their properties, computation methods, and relationships with helices, including specific cases on surfaces of revolution.

Contribution

It introduces the concept of isophote curves in Galilean space, distinguishes cases based on the nature of the axis vector, and provides methods for their computation on surfaces of revolution.

Findings

Derived formulas for isophote curves on surfaces of revolution

Established relationship between isophote curves and slant helices

Provided an example of computing isophote curves on isotropic surfaces

Abstract

In this paper, we introduce isophote curves on surfaces in Galilean 3-space. Apart from the general concept of isophotes, we split our studies into two cases to get the axis d of isophote curves lying on a surface such that d is an isotropic or a non isotropic vector. We also give the method to compute isophote curves of surfaces of revolution. Subsequently, we show the relationship between isophote curves and slant(general) helices on surfaces of revolution obtained by revolving a curve by Euclidean rotations. Finally, we give an example to compute isophote curves on isotropic surfaces of revolution.

Equations

$$\frac{⟨n(s,t),d⟩}{\Vert n(s,t)\Vert }=\cos \beta ,$$

$$\frac{⟨n(s,t),d⟩}{\Vert n(s,t)\Vert }=\cos \frac{\pi }{2}=0,$$

$$\bar{x}$$

$$\bar{y}$$

$$\bar{z}$$

$$⟨x,y⟩=\{\begin{array}{ll}{x}_1{x}_2,& \text{if }{x}_1\ne 0\text{or}{x}_2\ne 0,\\ y_1{y}_2+z_1{z}_2,& \text{if}{x}_1=0\text{and}{x}_2=0\end{array}$$

$$||x||=\{\begin{array}{ll}|x_1|,& \text{if }{x}_1\ne 0,\\ \sqrt{y_1^2+z_1^2},& \text{if}{x}_1=0.\end{array}$$

$$x\times y=\left|\begin{array}{ccc}0& e_2& e_3\\ x_1& y_1& z_1\\ x_2& y_2& z_2\end{array}\right|,$$

$$\alpha (s)=\left(s,f(s),g(s)\right).$$

$$T(s)$$

$$N(s)$$

$$B(s)$$

$$\kappa \left(s\right)$$

$$\tau \left(s\right)$$

$$T^{\prime }$$

$$N^{\prime }$$

$$B^{\prime }$$

$$\mathbf{X}(u_1,u_2)=(x(u_1,u_2),y(u_1,u_2),z(u_1,u_2)).$$

$$d{s}^2=\left(\begin{array}{cc}d{s}_1^2& 0\\ 0& d{s}_2^2\end{array}\right),$$

$$n=\frac{1}{\omega }(0,x_{u_2}{z}_{u_1}-x_{u_1}{z}_{u_2},x_{u_1}{y}_{u_2}-x_{u_2}{y}_{u_1}),$$

$$\omega =\sqrt{(x_{u_2}{z}_{u_1}-x_{u_1}{z}_{u_2}{)}^2+(x_{u_1}{y}_{u_2}-x_{u_2}{y}_{u_1}{)}^2}.$$

$$T^{\prime }$$

$$Q^{\prime }$$

$$n^{\prime }$$

$${\kappa }^2$$

$$k_g$$

$$Q$$

$$n$$

$$\vartheta =\sqrt{(y_2-x_2{)}^2+(y_3-x_3{)}^2}.$$

$$\phi =\frac{x_2{y}_2+x_3{y}_3}{\sqrt{y_2{}^2+y_3^2}}.$$

$$\cos \theta =\frac{x_2{y}_2+x_3{y}_3}{\sqrt{x_2{}^2+x_3^2}\sqrt{y_2{}^2+y_3^2}}.$$

$$⟨n,d⟩=\cos \theta =\text{constant}.$$

$$k_g⟨Q,d⟩+k_n⟨n,d⟩=0,$$

$$⟨Q,d⟩=-\frac{k_n}{k_g}\cos \theta .$$

$${\tau }_g⟨Q,d⟩=0,$$

$$d=\cos \theta n,$$

Full text

${\dagger}$ Department of Mathematics

Fırat University

23119 Elazig, Turkey

E-mail address: zuhal2387@yahoo.com.tr

${\ddagger}$ Department of Mathematics Education and RINS

Gyeongsang National University

Jinju 52828, Republic of Korea

E-mail address: dwyoon@gnu.ac.kr

†† Corresponding author: Dae Won Yoon.††2010 AMS Mathematics Subject Classification: 53A35, 53Z05.†† Key words and phrases: Galilean space, isophote curve, surfaces of revolution. The second author was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education(NRF-2018R1D1A1B07046979).

1 Introduction

The isophote curve method is one of the most efficient methods that can be used to analyze and visualize surfaces by lines of equal light intensity. Isophote curve whose normal vectors make a constant angle with a fixed vector(the axis) is one of the curves to characterize surfaces such as parameter, geodesics and asymptotic curves or lines of curvature. Moreover, this curve is used in computer graphics and it is also interesting to study for geometry.

The isophote curve of a given surface is calculated with two steps: firstly the normal vector field $n(s,t)$ of the surface is computed, and secondly the surface point is traced as

[TABLE]

where $\beta$ is a constant angle($0\leq\beta\leq\frac{\pi}{2}$).

Isophote curve is called a silhouette curve when the angle $\beta$ is given as a right angle such that

[TABLE]

where $d$ is the fixed vector.

From past to present, there have been a lot of researchers about isophote curves and their characterizations in [3, 4, 6, 7].

In this paper, our aim is to investigate isophote curves on surfaces in Galilean space and find its axis $d$ such that it is an isotropic and a non isotropic vector by means of the Galilean Darboux frame. According to the axis $d$, we split our studies into two cases to find the axis of isophote curves lying on a surface in Galilean space. Moreover, we give the method to compute isophote curves of surfaces of revolution obtained by revolving a curve by Euclidean and isotropic rotations.

2 Preliminaries

In accordance with the Erlangen Program, due to F. Klein, each geometry is associated with a group of transformations, and hence there are as many geometries as groups of transformations. Associated with group of transformations that in physics guarantees the invariance of many mechanical systems, the Galilei group, is the so-called Galilean geometry. That is, Galilean geometry is one of the nine Cayley-Klein geometries with projective signature $(0,0,+,+)$. The absolute of the Galilean geometry is an ordered triple $\{\omega,f,I\}$, where $\omega$ is the ideal (absolute) plane, $f$ the line in $\omega$ and $I$ the fixed elliptic involution of $f$.

We introduce homogeneous coordinates in ${G}_{3}$ in such a way that the absolute plane $\omega$ is given by $x_{0}=0$, the absolute line $f$ by $x_{0}=x_{1}=0$ and the elliptic involution by $(0:0:x_{2}:x_{3})\rightarrow(0:0:x_{3}:-x_{2})$.

The group of motions of $G_{3}$ is a six-parameter group given (in affine coordinates) by

[TABLE]

A plane is called Euclidean if it contains $f$, otherwise it is called isotropic or i.e., planes $x=consant$ are Euclidean, and so is the plane $\omega$. Other planes are isotropic. In other words, an isotropic plane does not involve any isotropic direction.

A Galilean scalar product of two vectors $x=(x_{1},y_{1},z_{1})$ and $y=(x_{2},y_{2},z_{2})$ in the Galilean 3-space $G_{3}$ is defined as

[TABLE]

and a Galilean norm of $x$ is given by

[TABLE]

A Galilean cross product of $x$ and $y$ on $G_{3}$ is defined by

[TABLE]

where $e_{2}=(0,1,0)$ and $e_{3}=(0,0,1)$, [8, 10].

Let $\alpha$ be an admissible curve of the class ${C}^{\infty}$ in ${G}_{3},$ and parametrized by the invariant parameter $s$, defined by

[TABLE]

Then the Frenet frame fields of $\alpha(s)$ are given by

[TABLE]

where the curvature $\kappa(s)$ and the torsion $\tau(s)$ of $\alpha(s)$ are written as, respectively,

[TABLE]

Here $T,$ $N$ and $B$ are said to be the tangent, principal normal and binormal vectors of $\alpha(s)$.

On the other hand, the Frenet formula of the curve is given by (cf. [9])

[TABLE]

Consider a $C^{r}$-regular surface $M$, $r\geq 1$, in $G_{3}$ parameterized by

[TABLE]

We denote by $x_{u_{i}}$, $y_{u_{i}}$ and $z_{u_{i}}$ the partial derivatives of the functions $x$, $y$ and $z$ with respect to $u_{i}$ ($i=1,2$), respectively.

On the other hand, the matrix of the first fundamental form $ds^{2}$ of a surface $M$ in $G_{3}$ is given by

[TABLE]

where $ds_{1}^{2}=(g_{1}du_{1}+g_{2}du_{2})^{2}$ and $ds_{2}^{2}=h_{11}du_{1}^{2}+2h_{12}du_{1}du_{2}+h_{22}du_{2}^{2}$. Here $g_{i}=x_{u_{i}}$ and $h_{ij}=\langle\tilde{\mathbf{X}}_{u_{i}},\tilde{\mathbf{X}}_{u_{j}}\rangle$ $(i,j=1,2)$ means the Euclidean scalar product of the projections $\tilde{\mathbf{X}}_{u_{i}}$ of vectors $\mathbf{X}_{u_{i}}$ onto the $yz$-plane.

The unit normal vector field $n$ of a surface $M$ is defined by

[TABLE]

where the positive function $\omega$ is given by

[TABLE]

Let $\{T,Q,n\}$ be a Galilean Darboux frame of $\alpha(s)$ with $T$ as the tangent vector of a curve $\alpha(s)$ in $G_{3}$ and $n$ be the unit normal to a surface and $Q=n\times T$. Then the Galilean Darboux frame is expressed as

[TABLE]

where $k_{g}$, $k_{n}$ and $\tau_{g}$ are the geodesic curvature, normal curvature and geodesic torsion of $\alpha(s)$ on $M$, respectively. Also, (2.2) implies

[TABLE]

where $\phi$ is an angle between the surface normal vector $n$ and the binormal vector $B$ of $\alpha,$ ([12]). A curve $\alpha(s)$ is a geodesic (an asymptotic curve or a line of curvature) if and only if $k_{g}$ ( $k_{n}$ or $\tau_{g}$) vanishes, respectively.

On the other hand, the usual transformation between the Galilean Frenet frames and the Darboux frames takes the form

[TABLE]

Artykbaev was introduced an angle between two vectors in Galilean space as follows:

Definition 2.1**.**

([1]) Let $x=(1,x_{2},x_{3})$ and $y=(1,y_{2},y_{3})$ be two unit non-isotropic vectors in $G{{}_{3}}$. Then an angle $\vartheta$ between $x$ and $y$ is defined by

[TABLE]

Definition 2.2**.**

([1]) An angle between a unit non-isotropic vector $x=(1,x_{2},x_{3})$ and an isotropic vector $y=(0,y_{2},y_{3})$ in $G{{}_{3}}$ is defined by

[TABLE]

Definition 2.3**.**

([1]) An angle $\theta$ between two isotropic vectors $x=(0,x_{2},x_{3})$ and $y=(0,y_{2},y_{3})$ parallel to the Euclidean plane in $G{{}_{3}}$ is equal to the Euclidean angle between them. That is,

[TABLE]

3 The axis of an isophote curve in Galilean Space

The starting point of this section is to get the fixed vector $d$ of an isophote curve via its Galilean Darboux frame.

Let $M$ be an admissible regular surface and $\alpha:I\subset R\rightarrow M$ be an unit speed curve parametrized by $\alpha(s)=(s,\alpha_{2}(s),\alpha_{3}(s))$ as an isophote curve for some $s\in I$.

In order to prove the results, we split it into two cases according to the fixed vector $d$.

Case 1. $d$ is an unit isotropic vector.

Since $n$ is the unit isotropic normal vector of a surface $M$, we have

[TABLE]

If we differentiate $\langle T,d\rangle=0$ with respect to $s$, using the Galilean Darboux frame (2.2), then we obtain

[TABLE]

which implies

[TABLE]

Taking account of the derivative of (3.1) we get

[TABLE]

where if  $\left\langle Q,d\right\rangle=0$, $k_{n}=0$ which means that $\alpha$ should be an asymptotic curve or $\tau_{g}=0$ which means that $\alpha$ should be a line of curvature. Then, for $k_{n}=0,$ $d$ can be written as

[TABLE]

since $d$ is a constant vector, $\tau_{g}$ should be equal zero. Also this is the trivial result.

For $\tau_{g}=0,d$ can be written as

[TABLE]

Since $\left\|d\right\|=1$, we get

[TABLE]

In this situation, we conclude that $\phi=\pm\theta$ or $\phi=\pi\pm\theta.$

From (2.3) and (2.4) in terms of the Galilean Frenet frame, we get

[TABLE]

If we differentiate (3.6) using (3.7) and $\tau_{g}=0$, we get $d^{\prime}=0,$ that is, $d$ is a constant isotropic vector. From now on, we suppose if $\alpha$ is a unit-speed isophote curve, then $\alpha$ is also a line of curvature.

Theorem 3.1**.**

Let $\alpha$ be a unit-speed isophote curve on a surface $M$ in $G_{3}$ with a fixed unit isotropic vector $d$ as the axis of the isophote curve. In that case, we have the following:

$i)$* If $\alpha$ is a geodesic curve, then $\alpha$ is a straight line.*

$ii)$* If $\alpha$ is an asymptotic curve on $M$, then it is a plane curve, and the fixed vector $d$ is spanned by $B.$*

Proof.

$i)$ If $\alpha$ is a geodesic curve, then we have $k_{g}=0$ and so from (3.2) it follows that $k_{n}=0,$ also $\tau_{g}=0.$ By substituting $k_{g}$ and $k_{n}$ into (2.3), we get $\kappa=0$, that is, $\alpha$ is a straight line.

$ii)$ If $\alpha$ is an asymptotic curve, we have $k_{n}=0.$ From (2.3) and (3.8), we obtain that

[TABLE]

Also, by substituting $\tau_{g}=0$ and $k_{n}=0$ into (2.4), we get $\tau=0.$ It means that $\alpha$ is a plane curve. $\square$

Theorem 3.2**.**

Let $\alpha$ be a unit-speed isophote curve on a surface $M$ in $G_{3}$ with a fixed unit isotropic vector $d$ as the axis of the isophote curve. The axis $d$ is perpendicular to the principal normal line of $\alpha$ if and only if  either $\alpha$ is a straight line, or an asymptotic curve on $M$ with taking $\dfrac{k_{n}}{k_{g}}=\tan\theta$ or $\alpha$ is a curve with $\frac{k_{n}}{k_{g}}=-\tan\theta.$

Proof.

If $\alpha$ is a unit-speed isophote curve with $\frac{k_{n}}{k_{g}}=\tan\theta$, then from (3.8), we get

[TABLE]

from this equation, we have $k_{g}=0$ or $\sin\theta=0$.

If $k_{g}=0$ then, from Theorem 3.1, $\alpha$ is a straight line.

If $\sin\theta=0$, then $k_{n}=0,$ that is, $\alpha$ is an asymptotic curve.

If we take $\frac{k_{n}}{k_{g}}=-\tan\theta$, then we can easily get $\left\langle N,d\right\rangle=0.$ $\square$

Theorem 3.3**.**

Let $\alpha$ be a unit-speed isophote curve on a surface $M$ in $G_{3}$ with a fixed unit isotropic vector $d$ as the axis of the isophote curve. The axis $d$ is perpendicular to the principal binormal line of $\alpha$ such that $\dfrac{k_{n}}{k_{g}}=\tan\theta$ if and only if  $\theta$ equals $\dfrac{\pi}{4}$.

Proof.

If $\alpha$ is a unit-speed isophote curve with $\frac{k_{n}}{k_{g}}=\tan\theta$, then from (3.8), we get

[TABLE]

Since $\alpha$ is a non-geodesic curve, $-\sin^{2}\theta+\cos^{2}\theta=0.$ So, $\tan\theta=1$. We know that $0\leq\theta\leq\dfrac{\pi}{2},$ then we get $\theta=\dfrac{\pi}{4}.$ $\square$

Theorem 3.4**.**

If $\alpha$ is a silhouette curve on $M,$ and $d$ is a unit isotropic vector such that it is parallel to $Q$, then the curve $\alpha$ is a plane curve.

Proof.

If a fixed vector $d$ is a unit isotropic vector and is parallel to $Q$, then we have

[TABLE]

By differentiating above equations with respect to $s$, we obtain

[TABLE]

Since $\alpha$ is a silhouette curve with $\langle n,d\rangle=0$, we get

[TABLE]

from this, we have $\tau=0.$ It means that $\alpha$ is a plane curve. $\square$

Case 2. Now, our aim is to find a fixed unit non-isotropic vector $d$ as the axis of an isophote curve.

Since $n$ is the unit isotropic normal vector of a surface $M$, we have

[TABLE]

Let $\alpha$ be a unit speed admissible isophote curve. If we differentiate

[TABLE]

with respect to $s,$ using the Galilean Darboux frame (2.2) then we have

[TABLE]

It follows from (2.6) that we find

[TABLE]

Taking account of the derivative of $\langle n,d\rangle=\varphi$ and using the Galilean Darboux frame (2.2)

[TABLE]

where if  $\left\langle Q,d\right\rangle=0$, then from (3.12) we get $k_{n}=0$ which means that  $\alpha$ should be an asymptotic curve. Then, for $k_{n}=0,$ $d$ can be written as

[TABLE]

Since $d$ is a constant vector, $k_{g}=$ $\varphi\tau_{g}$. Thus, we have the following result:

Corollary 3.5**.**

Let $\alpha$ be a unit-speed isophote curve on a surface $M$ in $G_{3}$ with a fixed unit non-isotropic vector $d$ as the axis of the isophote curve. If $\alpha$ is a geodesic curve or a line of curvature, then $\alpha$ is a straight line.

If $\tau_{g}=0$, that is, $\alpha$ is a line of curvature, then $d$ can be written as

[TABLE]

Since $d$ is a constant vector, $k_{g}=$ $k_{n}=0$, which implies $\kappa=0$, that is, $\alpha$ is a straight line.

Theorem 3.6**.**

Let $\alpha$ be a silhouette curve on $M$ and $d$ be a unit non-isotropic vector.

$i)$* If $d$ lies in the plane spanned by $T$ and $Q$, then $\alpha$ is a plane curve.*

$ii)$* If the axis $d$ is spanned by $T$, then $\alpha$ is a geodesic curve.*

Proof.

$i)$ Since $\alpha$ is a silhouette curve and $d$ is a unit non-isotropic vector, we get

[TABLE]

If we differentiate (3.16) with respect to $s$, then we get

[TABLE]

Since $d$ is lied in the plane spanned by $T$ and $Q,$ we get $k_{g}=0$. Also, if we differentiate $\left\langle n,d\right\rangle=0$ with respect to $s$, we get

[TABLE]

it follows that $\tau_{g}=0.$

Also, by substituting $\tau_{g}=0$ and $k_{g}=0$ into (2.3), we get $\tau=0.$ Thus, $\alpha$ is a plane curve.

$ii)$ If $d$ is spanned by $T$, then we get

[TABLE]

If we differentiate above equation, then $d^{\prime}=k_{g}Q$, it follows that $k_{g}=0$, that is, the curve is a geodesic curve. $\square$

4 Applications for Isophote Curves

We investigate an isophote curve among surfaces in Galilean space. Now we give some examples for this subject. To see this, notice that in $G_{3}$ surfaces of revolution are obtained by revolving a curve by Euclidean or isotropic rotations as follows, respectively,

[TABLE]

where $t$ is the Euclidean angle and

[TABLE]

where $t\in\mathbb{R}$ and $c=\text{constant}>0$.

The trajectory of a single point under a Euclidean rotation is a Euclidean circle

[TABLE]

The invariant $r$ is the radius of the circle. Euclidean circles intersect the absolute line $f$ in the fixed points of the elliptic involution $(F_{1},F_{2})$.

The trajectory of a point under isotropic rotation is an isotropic circle whose normal form is

[TABLE]

The invariant $c$ is the radius of the circle. The fixed line of the isotropic rotation is the absolute line f\[11]. For some more studies, see [2, 5].

If a curve $\alpha(s)=(f(s),0,g(s)),$ $\left(g(s)>0\right)$ is rotated by Euclidean rotations, then a surface of revolution is parametrized by

[TABLE]

If a curve $\alpha(s)$ is parametrized by the arc-length, then we take $f(s)=s.$ Then, the unit isotropic normal vector field $n(s,t)$ of $S$ is defined by

[TABLE]

where $S_{s}$ and $S_{t}$ are the partial differentiations with respect to $s$ and $t,$ respectively. Then, the isotropic normal vector is given by

[TABLE]

it becomes in terms of the Frenet frame as follows:

[TABLE]

Proposition 4.1**.**

Let a curve $\alpha(s)$ be a general helix with the isotropic axis $d$. Then, for $t_{0}=(\frac{2k+1}{2})\pi$ $(k\in Z),$ the curve $\alpha(s)$ on* surfaces of revolution given by (4.3) of revolution is an isophote curve with the axis $d$.*

Proof.

Substituting $t_{0}$ into (4.5), we get

[TABLE]

If $\alpha(s)$ is a general helix with the axis $d$, then $\left\langle B,d\right\rangle=$constant. Therefore, we get

[TABLE]

Thus $\alpha(s)$ is an isophote curve with the axis $d$ on the surfaces of revolution. $\square$

Proposition 4.2**.**

Let a curve $\alpha(s)$ be a slant helix with the isotropic axis $d$. Then, for $t_{0}=k\pi$ $(k\in Z),$ the curve $\alpha(s)$ on surfaces of revolution given by (4.3) is an isophote curve with the axis $d$.

Proof.

Substituting $t_{0}$ into (4.5), we get

[TABLE]

If $\alpha(s)$ is a slant helix with the axis $d$, then $\left\langle N,d\right\rangle=$constant. Therefore, we get

[TABLE]

Thus $\alpha(s)$ is an isophote curve with the axis $d$ on the surfaces of revolution. $\square$

If a curve $\alpha(s)=(f(s),0,g(s)),$ $\left(g(s)>0\right)$ is rotated by isotropic rotations, then a surface of revolution is parametrized by

[TABLE]

If a curve $\alpha(s)$ is parametrized by the arc-length, then we take $f(s)=s.$ Then, the isotropic surface normal is given by

[TABLE]

it becomes in terms of the Frenet frame as follows:

[TABLE]

Proposition 4.3**.**

*Let an isotropic axis $d$ is given by (0,$\mathit{d}_{y},d_{z}$). *

$i)$ If $\mathit{d}_{y}=0$ and $g(s)$ is a second order function, then the curve $\alpha(s)$ on surfaces of revolution given by (4.6) is an isophote curve.

$ii)$ If $\mathit{d}_{z}=0$ and $g(s)$ is a second order function, then the curve $\alpha(s)$* on surfaces of revolution given by (4.6) is an isophote curve.*

Proof.

$i)$ If $\mathit{d}_{y}=0,$* then we get $d=\lambda_{1}N,(\lambda_{1}\in R_{0})$*.

Using this above condition on (4.7), we get

[TABLE]

From the above equation, we can get $g(s)=\frac{s^{2}}{2c}+A,$ $A\in R.$ Thus we obtain $\left\langle n,d\right\rangle=\frac{\lambda_{1}}{\sqrt{2}}.$

$ii)$ If $\mathit{d}_{z}=0,$* then we get $d=-\lambda_{2}B,(\lambda_{2}\in R_{0}).$* Using this above condition on (4.7), we get

[TABLE]

From the above equation, we can get $g(s)=\frac{s^{2}}{2c}+A,$ $A\in R.$ Thus we obtain $\left\langle n,d\right\rangle=\frac{\lambda_{2}}{\sqrt{2}}.$ $\square$

Therefore, the rotating curve is an isotropic circle on surfaces of revolution. We also show the surfaces (4.6) for $g(s)=\frac{s^{2}}{2c}+A$ in Figure 1.

Corollary 4.4**.**

The generating curve $\alpha(s)=(f(s),0,g(s))$ on surfaces of revolution given by (4.6) becomes both a general helix and a slant helix with the axis $d$.