Geodesics with a normal distribution

Question

I'm working with the Desmos 3D calculator and I want to find the geodesic across some manifold currently I have a generalized equation for a generalized 3D distribution curve: $$f(x, y) = ab^{-{\left(\left(x-x_{1}\right)^{2}+\left(y-y_{1}\right)^{2}\right) \over c}} $$ with external variables $a, b, c, x_1,$ and $y_1$. I want know how I would find or approximate the geodesic of this graph (or any other manifold). I am currently trying to obtain the distance between some point $p$ and another point $p_1$ on our function $f(x,y)$ where the distance between $p$ and $p_1$ the distance is as small as possible. I plan to use this to iterate point $p$ by increasing it's $x$ and $y$ values by some amount, finding point $p_1$, moving point $p$ to point $p_1$,taking $p$'s last position and using it as a vector to find it's next position, increasing it by this vector, and then restarting from finding point $p_1$ though I don't know if this will work. any and all help is greatly appreciated, and I will provide a link to the graph I currently have if you need further understanding.

Answer 1

As clarified in the comments, the question answered below is "How can I plot geodesics on a surface in three-space, assuming we have a parametrization?"

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Here's a sketch that follows standard development of a differential geometry course. A bare-bones summary may be found at the bottom. It may be worth reading that first.

Page numbers and notation match Ted Shifrin's Differential Geometry Open Math Notes dated April 2021.

Parametrization

$\newcommand{\Mapx}{\mathbf{x}}\newcommand{\Nml}{\mathbf{n}}$(Chapter 2, pp. 35 ff.) Let $(u, v)$ denote plane coordinates in some region, assume $x$, $y$, and $z$ are smooth, real-valued functions in this region, and write $\Mapx = (x, y, z)$ for the resulting surface parametrization. (We'll assume $\Mapx$ satisfies a regularity hypothesis that's convenient to defer briefly.)

That is, the surface we're starting from is parametrized by $$ \Mapx(u, v) = \bigl(x(u, v), y(u, v), z(u, v)\bigr). $$ For a graph, we may use a Monge patch, i.e., where $\Mapx(u, v) = \bigl(u, v, \phi(u, v)\bigr)$ for some function $\phi$.

The vector-valued partial derivatives \begin{align*} \Mapx_{u} &= (x_{u}, y_{u}, z_{u}), \\ \Mapx_{v} &= (x_{v}, y_{v}, z_{v}) \end{align*} are tangent to the image of $\Mapx$. The regularity condition mentioned earlier is, at each point these vectors are linearly independent. In other words, the cross product $\Mapx_{u} \times \Mapx_{v}$ is nowhere-vanishing. (This is automatic for a Monge patch.) We define the induced unit normal field (p. 39) by $$ \Nml := \frac{\Mapx_{u} \times \Mapx_{v}}{\lVert\Mapx_{u} \times \Mapx_{v}\rVert}. $$

Fundamental Forms

We define the components of the first fundamental form (p. 39) to be \begin{align*} E &= \Mapx_{u} \cdot \Mapx_{u} = x_{u}^{2} + y_{u}^{2} + z_{u}^{2}, \\ F &= \Mapx_{u} \cdot \Mapx_{v} = x_{u} x_{v} + y_{u} y_{v} + z_{u} z_{v}, \\ G &= \Mapx_{v} \cdot \Mapx_{v} = x_{v}^{2} + y_{v}^{2} + z_{v}^{2}. \end{align*} (People also call these metric components in the given coordinates $(u, v)$, and use the notation $g_{ij}$, with $E = g_{11}$, $F = g_{12}$, and $G = g_{22}$.)

The functions $E$, $F$, and $G$ are defined even if $\Mapx$ is not regular. We have $0 \leq EG - F^{2}$ by the Cauchy-Schwarz inequality. Lagrange's identity for the cross product takes the form $\lVert\Mapx_{u} \times \Mapx_{v}\rVert = \sqrt{EG - F^{2}}$. Consequently, regularity is equivalent to $0 < EG - F^{2}$, and if $\Mapx$ is regular we have $$ \Nml = \frac{(y_{u}z_{v} - z_{u}y_{v}, z_{u}x_{v} - x_{u}z_{v}, x_{u}y_{v} - y_{u}x_{v})}{\sqrt{EG - F^{2}}}. $$

We define the components of the second fundamental form (p. 46) to be \begin{align*} \ell &= \Mapx_{uu} \cdot \Nml, \\ m &= \Mapx_{uv} \cdot \Nml, \\ n &= \Mapx_{vv} \cdot \Nml. \end{align*}

Christoffel Symbols

At each point of our surface, the vectors $\{\Mapx_{u}, \Mapx_{v}, \Nml\}$ are a basis of three-space. Consequently, every vector, such as $\Mapx_{uu}$ or $\Mapx_{uv}$ or $\Mapx_{vv}$, can be expressed as a linear combination of these basis elements. The normal components of the second derivatives are components of the second fundamental form. The tangential components are the Christoffel symbols (p. 57), defined by \begin{alignat*}{3} \Mapx_{uu} &= \Gamma_{uu}^{u} \Mapx_{u} &&+ \Gamma_{uu}^{v} \Mapx_{v} &&+ \ell \Nml, \\ \Mapx_{uv} &= \Gamma_{uv}^{u} \Mapx_{u} &&+ \Gamma_{uv}^{v} \Mapx_{v} &&+ m \Nml, \\ \Mapx_{vv} &= \Gamma_{vv}^{u} \Mapx_{u} &&+ \Gamma_{vv}^{v} \Mapx_{v} &&+ n \Nml. \end{alignat*} The three matrix equations on p. 58 (double dagger symbol, see also below) express the Christoffel symbols in terms of the first fundamental form components and their first partials.

Geodesic Equations

Every curve on our surface is the image of a curve $\bigl(u(t), v(t)\bigr)$ in the coordinate domain. A curve is a geodesic if and only if (p. 71, double clubsuit symbol) it satisfies the second-order non-linear ODE system \begin{alignat*}{4} u''(t) &+ \Gamma_{uu}^{u} u'(t)^{2} &&+ 2\Gamma_{uv}^{u} u'(t) v'(t) &&+ \Gamma_{vv}^{u} v'(t)^{2} && = 0, \\ v''(t) &+ \Gamma_{uu}^{v} u'(t)^{2} &&+ 2\Gamma_{uv}^{v} u'(t) v'(t) &&+ \Gamma_{vv}^{v} v'(t)^{2} && = 0. \end{alignat*}

Summary

Following this strategy, the task is in essence to

1. Calculate the Christoffel symbols in terms of the function $f$ whose graph is your surface of interest.
2. Solve the geodesic equations numerically, generating lists of $(u, v)$ values.
3. Plug the $(u, v)$ values into $\Mapx$, obtaining geodesics as curves on your surface.

Our surface is parametrized by $$ \Mapx(u, v) = \bigl(x(u, v), y(u, v), z(u, v)\bigr). $$ Calculate the three metric components \begin{align*} E &= \Mapx_{u} \cdot \Mapx_{u} = x_{u}^{2} + y_{u}^{2} + z_{u}^{2}, \\ F &= \Mapx_{u} \cdot \Mapx_{v} = x_{u} x_{v} + y_{u} y_{v} + z_{u} z_{v}, \\ G &= \Mapx_{v} \cdot \Mapx_{v} = x_{v}^{2} + y_{v}^{2} + z_{v}^{2}. \end{align*} Calculate the six Christoffel symbols \begin{align*} \left[\begin{array}{@{}c@{}} \Gamma_{uu}^{u} \\ \Gamma_{uu}^{v} \\ \end{array}\right] &= \frac{1}{EG - F^{2}}\left[\begin{array}{@{}rr@{}} G & -F \\ -F & E \end{array}\right] \left[\begin{array}{@{}c@{}} \frac{1}{2}E_{u} \\ F_{u} - \frac{1}{2}E_{v} \\ \end{array}\right], \\ % \left[\begin{array}{@{}c@{}} \Gamma_{uv}^{u} \\ \Gamma_{uv}^{v} \\ \end{array}\right] &= \frac{1}{EG - F^{2}}\left[\begin{array}{@{}rr@{}} G & -F \\ -F & E \end{array}\right] \left[\begin{array}{@{}c@{}} \frac{1}{2}E_{v} \\ \frac{1}{2}G_{u} \\ \end{array}\right], \\ % \left[\begin{array}{@{}c@{}} \Gamma_{vv}^{u} \\ \Gamma_{vv}^{v} \\ \end{array}\right] &= \frac{1}{EG - F^{2}}\left[\begin{array}{@{}rr@{}} G & -F \\ -F & E \end{array}\right] \left[\begin{array}{@{}c@{}} F_{v} - \frac{1}{2}G_{u} \\ \frac{1}{2}G_{v} \\ \end{array}\right]. \end{align*}

For step 2., we can introduce formal parameters $(U, V) = (u', v')$. The second-order geodesic equations in two unknown functions $(u, v)$ become first-order equations in four unknown functions $(u, v, U, V)$: \begin{align*} \left[\begin{array}{@{}c@{}} u' \\ v' \\ U' \\ V' \\ \end{array}\right] &= \left[\begin{array}{@{}c@{}} U \\ V \\ u'' \\ v'' \\ \end{array}\right] \\ &= \left[\begin{array}{@{}c@{}} U \\ V \\ -\Gamma_{uu}^{u}(u, v) U^{2} - 2\Gamma_{uv}^{u}(u, v) UV - \Gamma_{vv}^{u}(u, v) V^{2} \\ -\Gamma_{uu}^{v}(u, v) U^{2} - 2\Gamma_{uv}^{v}(u, v) UV - \Gamma_{vv}^{v}(u, v) V^{2} \end{array}\right]. \end{align*} Euler's method suffices to solve numerically; Desmos may be able to numerically solve ODE systems natively?

Step 3 is just a matter of evaluating $\Mapx$ at a set of data points.