thehog said:
bikenrrd said:
thehog said:
Catwhoorg said:
Specific to Cycling, the track has been mentioned multiple times on the NBC streams as being "fast with longer straights".
Anyone with better knowledge than I possess, able to comment on the track layout?
That is a question of mathematics; Can a velodrome track of equal length have desperate sized lengths and curves if each curve is 180c? The answer is no. It’s not possible.
Seriously? You can if the curves are (half) ellipses rather than semi-circles. You might not be able to successfully ride the track, but you could certainly build them with longer or shorter straights.
No, still not possible to do. If the track is to remain the same distance and the curves are 180 degrees, the length of the straights cannot change. If you cut the curve in half (through the cross section of the curve) then the straights will become longer but so will the length of the track.
Per my original statement:
That is a question of mathematics; Can a velodrome track of equal length have desperate sized lengths and curves if each curve is 180c? The answer is no. It’s not possible.
I'm sorry, thehog, but this is incorrect. It's mathematically possible to create tracks that have the same total length (in this case 250 meters), but different corner and straights lengths, even if you were to assume the corners are semicircles (half-circles) (which isn't even required by UCI regulations, see 3.6.067-.068). That the corners have to be 180° matters not, every semicircle is 180°, but the perimeter only depends on the radius (half of the diameter):
with r as the radius:
Code:
Perimeter cirle = 2 * pi * r
Permiter semicircle = 1/2 * 2 * pi * r = pi * r
For the proof I'm going to assume the bends are semicircles, not ellipses or other shapes, and the track has to have a total length of 250 meters, as that will reduce the involved variables to one. However, the simple principle will hold for different track lengths and corner shapes.
This is the general shape of a track, with two bend (B1 & B2) and two straights (S1 & S2); as you can see, the straights are parallel, as per UCI regulation:
We know that the length of the track has to be 250m according to UCI regulations for Olympic games and world championships. That means that the total circumference of the track in the diagram above has to be 250m (this 'line' is measured 20 cm above the inner edge of the actual track). This circumference includes the perimeter of two equal semicircles (B1 & B2) and two equal straight lines (S1 & S1).
The formula to calculate the the perimeter of a semicircle is just the formula for the circumference/perimeter of a circle devided by two:
Code:
Perimeter semicircle = 1/2 * 2 * pi * r # With r being the radius
As we have two semicircles, B1 & B2 with an equal radius, the total length of track running through both corners combined is just:
Code:
Combined corners B1 + B2 = 2 * (1/2 * 2 * pi * r) = 2 * pi * r
# Two semicircles make a circle...
We also have two straights of length y, so combining them we get:
Code:
Total length of straights combined = 2 * y
Adding the straights to the bends we get the total track length:
Code:
Total length = Two bends + two straigths = ( 2 * pi * r ) + ( 2 * y ) # Parentheses for seperation
As we know the length of the track has to be 250 meters, we can know calculate the length of the straights as a function of the radius of the bends:
Code:
total length = two bends + two straights;
total length - two bends = two straights;
(total length - two bends) / 2) = one straight
or:
250 = ( 2 * pi * r ) + ( 2 * y )
250 - 2 * pi * r = 2 * y
(250 - 2 * pi * r) / 2 = y
So, the length of a single straight is equal to (250 - 2 * pi * r) / 2
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UCI regulations state that a category 1 track (mandatory for Olympic Games) have a length of 250 m and a bend radius between 19-25m (See 3.6.095). That means that the longest straigth possible (with the narrowest bends of 19m) is:
Code:
(250 - 2 * pi * 19) / 2 = ~65,31 m
The shortest, with the widest semicircle bend of 25, is:
Code:
(250 - 2 * pi * 25) / 2 = ~46,46 m
Now, the UCI regulations do not specify that the bends have to be semicircles, only that they have to be safe to at least 75km/h. If you make the bends elliptical, it's possible to create even longer straights.
The challenge of designing a "fast" velodrome is thus one of finding the perfect balance between straights and the shape of the bend.