In this article, our goal is to show that the geodesics for a two-dimensional Euclidean space are straight lines.

In Cartesian coordinates and in two dimensional space, as there is no *z* coordinates, the Euclidean line element there becomes:

*dl ^{2} = dx^{2} + dy^{2}*

Therefore, the corresponding metric is - we are using Latin indices as we are not working in spacetime:

We also know from our previous article Geodesic equation and Christoffel symbols that the geodesic equation can be written as

But the Proper time is clearly not a convenient parameter in the case of the propagation of photons (the proper time is not defined for massless particles)

We should better use a so called affine parameter λ, as per below:

In order to calculate the eight Christoffel symbols (2*2*2 in 2D space), we need to use the equation given in Christoffel symbols in terms of the metric tensor

But as the values of the metric are constant (equal to 0 or 1 as pointed out above), the partial derivatives* g _{ij,k}* = 0 for all values of

*i, j*and

*k*. Therefore

**Γ**

_{jk}^{i}= 0**for all values**and the geodesic equation simply becomes :

*i, j, k*The function ** x^{i} = aλ + b** where a and b are constants is obviously a solution to this equation as when derived twice it gives 0.

As we are using Cartesian coordinates where* x^{i} *equals

*and*

**x***, the above equation becomes then*

**y**Solving for λ gives:

which is **the equation of a straight line with gradient c/a and constant (ad-bc)/a**.