Biking uphill by halving the effort

(Basic trigonometry concepts applied to biking uphill)

Passo dello Stelvio in Italy is one of the highest passes in Europe and the peculiarity of the road design is someway related to the content of this story — Photo by David Dvořáček on Unsplash

I know that this article will not add any wondrous new engineering solutions, but sometimes even reviewing basic trigonometry concepts can be helpful in explaining everyday phenomena. For example, I was cycling uphill near my place, and everyone knows that pedaling along the line of maximum slope is more tiring than going in a zigzag path.

So I wondered two questions:

1. what the relationship is between the different angles that come into play in the context of biking uphill
2. what is the average deviation from the maximum slope angle I should turn my bike handlebar in order to half the uphill slope angle.

The concept of the problem can be illustrated as in the figure below that can be recreated using the following data:

Plan equation: y=2z; Points coordinates:A=(2,2,1);B=(2,0,0);C=(2,2,0) and Origin (0,0,0).

This is a representation of an inclined plane, our uphill, with the red line being the maximum slope line and the green line an alternative deviation from the red line. (Picture original of the author using Geogebra)

The red line represents the inclination of the plane or the uphill that we would have to travel if we proceeded along the maximum slope. The below red right triangle shows at the base the alpha angle that we assume in our example to be about 26º.

The green line instead represents a possible alternative path that we could take if we proceed with an uphill angle different from that of maximum slope. By the properties of triangles, this angle is represented here by Beta. (I could also have drawn the green line from B, but I would have ended up at a point A’ further to the left or I could also have drawn the zigzag path to end up exactly at point A but in the end…it’s the same thing)

The green triangle also has an angle at the base (gamma) which intuitively is smaller than alpha and it’s a function of alpha and beta. So the first question is to determine the relationship that links these 3 angles:

• Alpha (⍺): uphill angle of maximum slope line
• Beta (β): deviation angle from the maximum slope line (our bike handlebar)
• Gamma (𝛾): the resulting perceived uphill angle we should face in our path as a function of alpha and beta.

The trigonometric formulas binding the three angles can be deduced as:

AC=AB*sin(⍺)

but AC is also part of the AOC triangle so:

AC=OA*sin(𝛾)

From which of course because AC=AC:

AB*sin(⍺)=OA*sin(𝛾)

but:

AB = OA*cos(β) because the angle of triangle ABO is also right in B.

Substituting AB we have:

OA*cos(β)*sin(⍺) = OA*sin(𝛾)

OA disappears and therefore the final relationship that links the three angles is:

cos(β)*sin(⍺)=sin(γ).

So if alpha in our example is 26.5651º and we go zig zag deviating from maximum slope line of beta=41.8103º the actual angle we face is gamma=19.4712º which is a good decrease of the actual slope of about 27%.

Now let’s assume we want to answer our second question: which angle (beta) I should deviate from the max slope line in order to have an uphill angle gamma that’s half of the inclined plane angle alfa.

We have that alpha = 26.5651º, so we want to find beta such that gamma = 0.5 alpha = 13.28º. We can say in this case that:

⍺ = 2𝛾

In this case the above formula becomes:

cos(β)sin(2𝛾)=sin(𝛾)

but for another trigonometric property :

sin(2𝛾)=2sin(𝛾)cos(𝛾)

substituting:

2cos(β)sin(𝛾)cos(𝛾)=sin(𝛾)

2cos(β)cos(𝛾)=1

cos(β)=1/(2cos(𝛾))

β = arccos(1/(2cos(𝛾)))

This function is defined for gamma between -60º and +60º.

Picture below shows on X axis the perceived inclination 𝛾 of the plane we are biking (note: which is not the inclination angle of the actual plane ⍺ represented by the angle of the uphill along the maximum slope) and in the Y axis the angle of deviation of our bike handlebar from the max slope line to halve the max slope angle. So for example if our road has an angle of 10º (Note: Consider that the typical roads for cars and bikes are in the range of +-20% meaning that the angles of the inclined plan are within the ranges of +- arctan(20/100) = approx. +-10 º) and we want to half that angle (=> 5 º) then our handlebar should deviate from max slope line of (see the green rectangle) about 60º.

On the X axis the gamma angle (=1/2 alpha) while on the Y the deviation from the max slope line. This means that if I turn my handlebar to form a 60 degree angle with the max slope line of the uphill I am biking uphill at a perceived angle which is the half of the actual uphill angle. Consider that most of the roads are built with a maximum angle of +-10 degrees, so halving that means approximately being in the green rectangle. (Picture original of the author using Geogebra)

This means that regardless of the road you are on, if you want to reduce the the uphill angle of a half you should point your bike handlebar of 60 º left and right zigzagging uphill. In this way for sure you run a longer path but the effort required should be theoretically … half.

Have a good biking.