Home > Differential Geometry > Car movement – differential geometry application

Car movement – differential geometry application

The problem presented below is from my differential geometry course. The initial reference is Nelson, Tensor Analysis 1967. The car is modelled as follows:

Denote by C(x,y) the center of the back wheel line, \theta the angle of the direction of the car with the horizontal direction, \phi the angle made by the front wheels with the direction of the car and L the length of the car.

The possible movements of the car are denoted as follows:

  •  steering: S=\displaystyle\frac{\partial}{\partial \phi};
  •   drive: D=\displaystyle\cos \theta \frac{\partial}{\partial x}+\sin\theta \frac{\partial}{\partial y}+\frac{\tan \phi}{L}\frac{\partial}{\partial \theta};
  •  rotation: R=[S,D]=\displaystyle\frac{1}{L\cos^2 \phi}\frac{\partial }{\partial \theta};
  •   translation: T=[R,D]=\displaystyle\frac{\cos \theta}{L\cos^2 \phi}\frac{\partial}{\partial y}-\frac{\sin\theta}{L\cos^2\phi}\frac{\partial}{\partial x}

Where [X,Y]=XY-YX. All these transformations seem very logical. The interpretations are quite interesting:

– from the expression of D, when the car is shorter, you can change the orientation of the car very easily, but when it is longer, like a truck, you it is not that easy ( see the term with \frac{\partial}{\partial \theta})
– the rotation is faster for smaller cars, and for greater steering angle
– translation is easier for smaller cars.

This is not a problem. It’s just a nice application of differential geometry. This presentation can generate different problems. For example:

Everyone knows that it’s not easy for a beginner to do a lateral parking. Find the least number of necessary moves to do a lateral parking, using the things presented above.

For more details about this interpretation, you could see this thread on Math Overflow.


  1. Srinivas Rau
    June 2, 2011 at 3:22 pm

    Please state the problem!We have only the definitions here!

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