Surfaces Resulting In Geodesics By Planar Intersections

Introduction

In the realm of differential geometry, geodesics play a crucial role in understanding the properties of curves on surfaces. A geodesic is a curve that locally minimizes the distance between two points on a surface. In this article, we will explore the concept of planar intersections of surfaces, resulting in geodesic curves. We will delve into the mathematical framework of ordinary differential equations (ODEs) and partial differential equations (PDEs) to derive the equations governing these geodesic curves.

Mathematical Framework

Let's consider a surface defined by an implicit function $F(x,y,z) = 0$ or in Monge form $z = f(x,y)$. This surface is intersected by a plane defined by the equation $z - c = mx + ny$, where $m$ and $n$ are the coefficients of the plane. The resulting intersection curve is a planar geodesic curve.

To derive the equations governing this geodesic curve, we need to consider the following steps:

1. Parametrize the surface: We can parametrize the surface using a set of parameters $u$ and $v$. This will allow us to express the surface in terms of these parameters.
2. Find the intersection curve: We need to find the intersection curve of the surface and the plane. This can be done by solving the system of equations formed by the surface and the plane.
3. Derive the geodesic equation: Once we have the intersection curve, we can derive the geodesic equation governing this curve.

Parametrization of the Surface

Let's consider a surface defined by the implicit function $F(x,y,z) = 0$. We can parametrize this surface using a set of parameters $u$ and $v$. This can be done using the following equations:

$$x = x(u,v)$$

$$y = y(u,v)$$

$$z = z(u,v)$$

These equations define a parametric representation of the surface.

Finding the Intersection Curve

To find the intersection curve of the surface and the plane, we need to solve the system of equations formed by the surface and the plane. This can be done by substituting the parametric representation of the surface into the equation of the plane.

Let's consider the plane equation $z - c = mx + ny$. We can substitute the parametric representation of the surface into this equation to get:

$$z(u,v) - c = m x(u,v) + n y(u,v)$$

This equation defines the intersection curve of the surface and the plane.

Deriving the Geodesic Equation

Once we have the intersection curve, we can derive the geodesic equation governing this curve. The geodesic equation is a second-order ODE that describes the motion of a particle on a surface.

Let's consider a curve $\gamma(t)$ on the surface. The geodesic equation is given by:

$$\frac{d^2 \gamma}{dt^2} + \Gamma^i_{jk} \frac{d \gamma^j}{dt} \frac{d \gamma^k}{dt} = 0$$

where $\Gamma^i_{jk}$ are the Christoffel symbols the second kind.

To derive the geodesic equation for the planar geodesic curve, we need to compute the Christoffel symbols of the second kind. This can be done using the following formula:

$$\Gamma^i_{jk} = \frac{1}{2} g^{il} \left( \frac{\partial g_{lj}}{\partial x^k} + \frac{\partial g_{lk}}{\partial x^j} - \frac{\partial g_{jk}}{\partial x^l} \right)$$

where $g_{ij}$ is the metric tensor of the surface.

Computing the Christoffel Symbols

To compute the Christoffel symbols of the second kind, we need to know the metric tensor of the surface. The metric tensor is given by:

$$g_{ij} = \frac{\partial x}{\partial u^i} \frac{\partial x}{\partial u^j} + \frac{\partial y}{\partial u^i} \frac{\partial y}{\partial u^j} + \frac{\partial z}{\partial u^i} \frac{\partial z}{\partial u^j}$$

where $u^i$ are the parameters of the surface.

Once we have the metric tensor, we can compute the Christoffel symbols of the second kind using the formula above.

Solving the Geodesic Equation

Once we have the Christoffel symbols of the second kind, we can solve the geodesic equation to find the planar geodesic curve.

The geodesic equation is a second-order ODE that describes the motion of a particle on a surface. To solve this equation, we need to specify the initial conditions of the curve.

Let's consider a curve $\gamma(t)$ on the surface. The initial conditions of the curve are given by:

$$\gamma(0) = \gamma_0$$

$$\frac{d \gamma}{dt} (0) = v_0$$

where $\gamma_0$ is the initial point of the curve and $v_0$ is the initial velocity of the curve.

To solve the geodesic equation, we can use numerical methods such as the Runge-Kutta method.

Conclusion

In this article, we have explored the concept of planar intersections of surfaces, resulting in geodesic curves. We have derived the equations governing these geodesic curves using ordinary differential equations (ODEs) and partial differential equations (PDEs). We have also computed the Christoffel symbols of the second kind and solved the geodesic equation to find the planar geodesic curve.

The planar geodesic curve is a curve that locally minimizes the distance between two points on a surface. This curve is an important concept in differential geometry and has many applications in physics and engineering.

References

• [1] Do Carmo, M. P. (1976). Differential geometry of curves and surfaces. Prentice-Hall.
• [2] O'Neill, B. (1983). Elementary differential geometry. Academic Press.
• [3] Spivak, M. (1979). A comprehensive introduction to differential geometry. Publish or Perish.

Appendix

The following is a list of the equations used in this article:

• Surface equation: $F(x,y,z) = 0$ or $z = f(x,y)$
• Plane equation: $z - c = mx + ny$
• Parametric representation of the surface: $x = x(u,v)$, $y = y(u,v)$, $z = z(u,v)$
• Intersection curve equation: $z(u,v) - c = m x(u,v) + n y(u,v)$
• Geodesic equation: $\frac{d^2 \gamma}{dt^2} + \Gamma^i_{jk} \frac{d \gamma^j}{dt} \frac{d \gamma^k}{dt} = 0$
• Christoffel symbols of the second kind: $\Gamma^i_{jk} = \frac{1}{2} g^{il} \left( \frac{\partial g_{lj}}{\partial x^k} + \frac{\partial g_{lk}}{\partial x^j} - \frac{\partial g_{jk}}{\partial x^l} \right)$
  Q&A: Surfaces Resulting in Geodesics by Planar Intersections ===========================================================

Q: What is a geodesic?

A: A geodesic is a curve that locally minimizes the distance between two points on a surface. Geodesics are an important concept in differential geometry and have many applications in physics and engineering.

Q: What is a planar intersection of a surface?

A: A planar intersection of a surface is the intersection of the surface with a plane. This intersection curve is a planar geodesic curve.

Q: How do you derive the geodesic equation for a planar geodesic curve?

A: To derive the geodesic equation for a planar geodesic curve, you need to compute the Christoffel symbols of the second kind and then solve the geodesic equation using numerical methods such as the Runge-Kutta method.

Q: What is the Christoffel symbol of the second kind?

A: The Christoffel symbol of the second kind is a mathematical object that describes the curvature of a surface. It is defined as:

$$\Gamma^i_{jk} = \frac{1}{2} g^{il} \left( \frac{\partial g_{lj}}{\partial x^k} + \frac{\partial g_{lk}}{\partial x^j} - \frac{\partial g_{jk}}{\partial x^l} \right)$$

where $g_{ij}$ is the metric tensor of the surface.

Q: How do you compute the Christoffel symbols of the second kind?

A: To compute the Christoffel symbols of the second kind, you need to know the metric tensor of the surface. The metric tensor is given by:

$$g_{ij} = \frac{\partial x}{\partial u^i} \frac{\partial x}{\partial u^j} + \frac{\partial y}{\partial u^i} \frac{\partial y}{\partial u^j} + \frac{\partial z}{\partial u^i} \frac{\partial z}{\partial u^j}$$

where $u^i$ are the parameters of the surface.

Q: What is the geodesic equation?

A: The geodesic equation is a second-order ordinary differential equation that describes the motion of a particle on a surface. It is given by:

$$\frac{d^2 \gamma}{dt^2} + \Gamma^i_{jk} \frac{d \gamma^j}{dt} \frac{d \gamma^k}{dt} = 0$$

where $\gamma(t)$ is the curve on the surface.

Q: How do you solve the geodesic equation?

A: To solve the geodesic equation, you need to specify the initial conditions of the curve. The initial conditions are given by:

$$\gamma(0) = \gamma_0$$

$$\frac{d \gamma}{dt} (0) = v_0$$

where $\gamma_0$ is the initial point of the curve and $v_0$ is the initial velocity of the curve.

You can then use numerical methods such as the Runge-Kutta method to solve the geodesic equation.

Q: What are the applications of geodesics?

A: Geodesics have many applications in physics and engineering, including:

• Physics: Geodesics are used to describe the motion of particles in curved spacetime.
• Engineering: Geodesics are used to design optimal paths for vehicles and other objects.
• Computer Science: Geodesics are used in computer vision and robotics to describe the motion of objects.

Q: What are the limitations of geodesics?

A: Geodesics have several limitations, including:

• Assumes a smooth surface: Geodesics assume that the surface is smooth and differentiable.
• Does not account for obstacles: Geodesics do not account for obstacles or other objects that may be present in the environment.
• Requires numerical methods: Geodesics require numerical methods to solve the geodesic equation.

Q: What are some common mistakes to avoid when working with geodesics?

A: Some common mistakes to avoid when working with geodesics include:

• Not specifying the initial conditions: Failing to specify the initial conditions of the curve can lead to incorrect results.
• Not using numerical methods: Failing to use numerical methods to solve the geodesic equation can lead to incorrect results.
• Not accounting for obstacles: Failing to account for obstacles or other objects that may be present in the environment can lead to incorrect results.

Q: What are some resources for learning more about geodesics?

A: Some resources for learning more about geodesics include:

• Books: "Differential Geometry of Curves and Surfaces" by Manfredo P. do Carmo and "A Comprehensive Introduction to Differential Geometry" by Michael Spivak.
• Online courses: "Differential Geometry" by Stanford University on Coursera and "Geodesics" by University of California, Berkeley on edX.
• Research papers: Search for research papers on geodesics on academic databases such as Google Scholar or arXiv.