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What is geodesic distance? How is it used in mathematical morphology?
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Geodesic distance refers to the shortest path between two points on a curved surface or in a metric space. In simpler terms, it is the length of the shortest route that connects two points while adhering to the constraints of the space in which they reside. Show more…
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The shortest path between two points on a curved surface, such as the surface of a sphere, is called a geodesic. To find a geodesic, one has first to set up an integral that gives the length of a path on the surface in question. This will always be similar to the integral (6.2) but may be more complicated (depending on the nature of the surface) and may involve different coordinates than $x$ and $y$. To illustrate this, use spherical polar coordinates $(r, \theta, \phi)$ to show that the length of a path joining two points on a sphere of radius $R$ is $L=R \int_{\theta_{1}}^{\theta_{2}} \sqrt{1+\sin ^{2} \theta \phi^{\prime}(\theta)^{2}} d \theta$ if $\left(\theta_{1}, \phi_{1}\right)$ and $\left(\theta_{2}, \phi_{2}\right)$ specify the two points and we assume that the path is expressed as $\phi=\phi(\theta)$ (You will find how to minimize this length in Problem $6.16 .)$
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