Originally Posted by
Road Fan
The site is not wrong and it is not my site. It asks you for inputs that seem convenient: rim bead seat diameter spec (622, 630, etc) and tire width (22, 28, 42, etc). With those inputs the code can compute wheel radius in a code unit. The writer shows us how to input the numbers he needs rather than teaching the mathematics. The Wiki article "Bicycle and Motorcycle Geometry" uses radius, but it is trying to show the science.
Sorry for my typing, i'm in a cast due to a broken wrist.
I've taken bike specs from the literature together with my measurements, and run the numbers both on thw site and on the equation, validating against specs. If I use wheel radius, the site result will agree with the equation, with my measurements, and the published specs.
I'd suggest reading Chapter 8 of "Bicycling Science 3rd edition" by Wilson and Papadopous for a detailed discussion of what influences bike handling. It does not explain popular terms such as "twitchiness."
recap:
not my site
site is correct because it checks with my measurements, specs, and calculations
wheel radius is correct because it results in the agreement mentioned above
suggest delving into the Wiki site and the text "Bicycling Science."
What I can't steer you toward is a derivation of the formula based on pure geometry.
Are you talking about calculating trail from scratch? You don't need to reference a circle at all. You need to plot two right triangles, one with a leg that is rake and has the HTA in it to get the hypotenuse, then you take that hypotenuse an subtract it from the total hub to ground distance (total wheel radius), and use what's left and the angles provided to get the length of the leg parallel to the ground.
That's all trail is. You can make the calculation more involved if you want the formula to derive your total wheel radius from your rim radius and tire size, but you can also just get that off a chart or with a ruler.