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In my text book the definition of conformal mapping is given as follows:

Definition:An analytic mapping $f$ in a domain $\Omega$ is said to be conformal at $z_0\in\Omega$ if ,whenever $\gamma_1$ and $\gamma_2$ are parameterized curves intersecting at $z_0=\gamma_1(t_0)=\gamma_2(t_0)$ with non-zero tangents then the following hold:

(1)The two transfored curves $\Gamma_1=fo\gamma_1$ and $\Gamma_2=fo\gamma_2$ have non zero tangents at $t_0$.

(2)The angle from $\Gamma'_1(t_0)$ to $\Gamma'_2(t_0)$ is same as angle from $\gamma'_1(t_0)$ $\gamma'_2(t_0)$

My question: What is the harm if we take the tangent of curve at point $t_0$ to be zero,why it is necessary to take only those curve which has non zero tangents at that point?Even if one of these curve is has zero tangent(or both of them) then also we can find the direction between directed curves.

Thanks in advance,any help will be appriciable!

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    $\begingroup$ The direction is usually defined as $\arg \gamma'(t_0)$, how would you define that if $ \gamma'(t_0) = 0$? $\endgroup$
    – Martin R
    Sep 17, 2020 at 7:50

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The angle between two vectors $v,w$ is determined by the scalar product $\langle v/\vert v\vert,w/\vert w\vert\rangle$ via

$$\cos\alpha=\left\langle \frac{v}{\vert v \vert},\frac{w}{\vert w\vert}\right\rangle.$$

But this expression doesn't make any sense if $\vert v\vert$ or $\vert w\vert$ vanish. So to even talk about the angle between two vectors they need to be non-zero. And the angle between two curves is the angle between their tangent vectors, which should thus be non-zero, otherwise the angle isn't well-defined.

For instance, the curve $(t^3,t^2)$ has a sharp point at $t=0$, see this desmos graph. It is also differentiable, and it intersects the curve $(t,0)$ at $t=0$. But at what angle? The point of the former curve makes it impossible to pick a reasonable one, and that's because its derivative at $t=0$ vanishes. So we have to exclude such curves.

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The direction of a curve is given by the tangent vector.

Take the curve from $[-1;1]$ to $\mathbb{C}$ given by $t\mapsto (t^2,0)$. At $t \leq 0$ the curve go "from the right to the left" of the complexe plane. At $t \geq 0$ the curve go "backward". So at $0$ you can not define a direction.

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