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$$β=arctan [ (1+tan^2 \theta ) K\sin \alpha + tan \theta \sqrt {1+(1+tan^2 \theta )K^2 \sin^2 \alpha} ]$$

This is the general straight line (circle) equation on the sphere. $\alpha$ is longitude, $\beta$ is latitudes, $K$ is the slope of a straight line (circle) and $\theta $ is the distance between the point of the straight line and the equator when alpha equals zero.

diagram

Diagram, the horizontal line (circle) is the equator, and its $K=0$. The $K$ values of other lines (circles) are $-0.3$, and their $\theta $ values are $0.45pi, \ 0.25pi, \ 0, \ 0.$

The calculation formula of the slope of the spherical curve is:

$$K={\sinβ_2-\sinβ_1\over\sinα_2\cosβ_2-\sinα_1\cosβ_1}$$

In the sphere very small area, this formula is approximate to the formula of the slope of the plane curve. So the slope of a plane curve is only a special case of the slope of a spherical curve.

According to this formula, we can find the derivative of the spherical curve and the calculus operation for the sphere.

Reference https://sphericalparallelism.quora.com/When-the-diameter-of-the-sphere-tends-to-infinity-what-does-the-small-circle-tend-to?share=ce851a1c&srid=5i3fQ

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    $\begingroup$ Definitions and terminology can be improved. $\endgroup$ – Narasimham Mar 29 '18 at 19:11
  • $\begingroup$ @Narasimham Do you mean I want to improve? $\endgroup$ – enbin zheng Mar 29 '18 at 21:14
  • $\begingroup$ I find a bit confusing. About the reference link, great circles have zero slip (geodesics ) which is rate of chage of opening radius ( $r_o$ Clairaut's constant) with respect to axial distance z of For small circles slip $ \tan \gamma= d r_o/d z,\, \gamma$ is angle between sphere normal and local arc normal without sphere from Frenet Serret relations. $\endgroup$ – Narasimham Mar 30 '18 at 4:06
  • $\begingroup$ @Narasimham The link is to say that the curvature of a small circle is equal to zero. Because the slope of the latitudinal circle equals zero. $\endgroup$ – enbin zheng Mar 30 '18 at 5:41
  • $\begingroup$ I am not able to understand this statement. Small circles have geodesic curvature $ \kappa_g\ne 0$ , great circle geodesics have it vanish. Did you mean torsion of small circle in space $=0$ without reference to sphere? $\endgroup$ – Narasimham Mar 30 '18 at 6:33
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You could correlate with the following result that has a slightly different notation ($\varphi$ latitude, $\theta $ longitude, $ \psi $ angle small circle makes to longitude, $\alpha $ inclination of small circle to plane of equator).

$$ \tan \alpha = \dfrac{\sin \theta \cos \varphi + \cos \theta \tan \psi}{ \sin \varphi}{} $$

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  • $\begingroup$ $$\phi =arctan ( (1+tan^2 ψ ) \tan \alpha \sin \theta + tan ψ \sqrt {1+(1+tan^2 ψ ) \tan \alpha^2 \sin^2 \theta} ) $$Is that what it means? $\endgroup$ – enbin zheng Mar 29 '18 at 21:55
  • $\begingroup$ You brought in a new $\phi$ without defining it. Maybe you get better help with your derivation, not just the end result. $\endgroup$ – Narasimham Mar 30 '18 at 4:14
  • $\begingroup$ phi is latitudes. $\endgroup$ – enbin zheng Mar 30 '18 at 5:37
  • $\begingroup$ What does your formula mean, I don't understand.? $\endgroup$ – enbin zheng Mar 30 '18 at 20:55

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