Circles and Euler spiral (repost from general math)

In summary, the conversation discusses a problem involving two circles and finding the Euler spiral tangent points. The problem may be solved through iterative or analytical methods. The known and unknown data are listed, and a document by Professor R.E. Deakin is referenced for further information. The poster expresses gratitude for any help with the problem.
  • #1
Mario
2
0
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...
XjDbYHOr_Br-kCIA9dbKtPZ3_hF3oLd2wvA82IBgzcQ?dl=0&size=1280x960&size_mode=3.png
 
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  • #2
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 ?
 
  • #3

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
E0SKIm4P2938lm3JMCH_vwOsLQWlWuyP7WGDQ5v-mJg?dl=0&size=2048x1536&size_mode=3.png

RWiXdEsMMiiWUQ1ILFSpS2yKHG1QW3iBUXhgVIgtWOc?dl=0&size=2048x1536&size_mode=3.png

k7ROPOg-Qfh8T1uy89Tmp0PlB_lKnOLIdIQ1rLqi_z0?dl=0&size=1600x1200&size_mode=3.png

HdVyqmRSCaP-oZm8RR8FCKFmWVrehBDMS5t_4caiz1s?dl=0&size=1600x1200&size_mode=3.png

(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
 

Related to Circles and Euler spiral (repost from general math)

1. What is the difference between a circle and an Euler spiral?

A circle is a geometric shape with a constant radius and a circumference that is always the same distance from the center. An Euler spiral, also known as a clothoid or Cornu spiral, is a curve that gradually increases in curvature. It is often used in engineering and architecture for its smooth transition between straight and curved sections.

2. How is an Euler spiral related to calculus?

An Euler spiral is a mathematical curve that can be described using the Fresnel integral, which involves the use of calculus. The shape of the curve is determined by the rate of change of curvature at each point, making it a useful concept in calculus and differential geometry.

3. What is the practical application of an Euler spiral?

Euler spirals have many practical applications, such as in road design, where they are used to create smooth curves for highways and ramps. They are also used in the design of roller coaster tracks, where they help to create a thrilling and smooth ride. In addition, they have applications in optics, where they are used to describe the shape of wavefronts.

4. Can an Euler spiral be approximated by a series of straight lines?

Yes, an Euler spiral can be approximated by a series of straight lines, but the accuracy of the approximation depends on the number of lines used. The more lines used, the closer the approximation will be to the actual curve. This is known as the polygonal approximation method.

5. Who discovered the Euler spiral?

The Euler spiral was first described by Swiss mathematician Leonhard Euler in the 18th century. It was later studied and popularized by French mathematician Marie Alfred Cornu in the 19th century, leading to its alternative name of the Cornu spiral.

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