Circles and Euler spiral (repost from general math)

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Mario
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Hi,
i have this problem..., giving two circle (example radius 1 = 500 units, radius 2 = 200 units, distance between centers = 275.73 units) find Euler Spiral (aka Cornu spiral, aka Clothoid https://en.wikipedia.org/wiki/Euler_spiral) tangent giving circle (unknown tangent points).
For this problem are two mirrored Euler spiral as solution with length = 450 units.
Problem so simply to explain but not so simply to find solution...
Many thanks for help...
CAD drawing of a large yellow circle labelled R500 and a smaller cyan circle labelled R200, centres offset vertically by 275.73, with a magenta and blue arc bridging their lower tangent region and a dimension labelled 450 along the arc.
 

Discussion

Hello again, Mario, and :welcome: again, too

Well, we are making progress. There is a problem statement and the link gives a bunch of equations. So we can claim 1 and 2 below are filled in.
All we have to do now is make a start with 3 !

Homework Statement

Homework Equations

The Attempt at a Solution


[/B]
Oh, and: don't erase the template (or was your thread moved here by a moderator?)

For a start: how did you find the 275.73 ?
 

Homework Statement


[/B]
known data:
center of circle 1 (xO1,yO1) and radius of circle = R1
center of circle 2 (xO2,yO2) and radius of circle = R2

unknown data:
lenght of spiral (magenta and blue) L=?
tangent point of spiral on circle 1 (xA, yA) = ?
tangent point of spiral on circle 2 (xB, yB) = ?

Homework Equations



Particular declination of problem are resolved with below equation, but in this resolution know data are:
R1, R2, L, xA=zero, yA=zero, xO1=zero, yO1=R1, line(O1-A) is orthogonal to x axes
calculated data are:
xO2, yO2, xB, yB
Geometry diagram of an Euler spiral transition curve joining two circles: circle 1 centred at O1 with radius R1 and circle 2 centred at O2 with radius R2, with tangent points A and B, arc length s and L marked, a point P on the spiral, radius rho, and tangent angle phi and phi_B with x and y axes.

Section heading 5.8 'Clothoid transition curves between circular curves' followed by a sentence defining a clothoid transition joining circles of radii R1 and R2, with curvature varying from 1/R1 to 1/R2.

Textbook page showing Figure 5.6 description of a clothoid transition curve AB joining two circular arcs of radii R1 and R2, followed by equations 5.34 and 5.35 for curvature and elemental arc length, and integration to the tangential angle.

Typeset derivation of Euler spiral (clothoid) equations: tangent angle phi = s^2/(2R1) type expressions, with x and y coordinates given as series expansions in s, labelled equations 5.36 to 5.42.

(complete document at this link http://www.mygeodesy.id.au/documents/Horizontal Curves.pdf prof. R.E.Deakin)

The Attempt at a Solution



A iterative solution is using the above equations and change L to find desidered distance of two circle...
but Analitical solution is possible ?
many thanks

P.S. move post here, if I understood well, it was a hint