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## 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.