Relation between canonical forms and volume of polytopes

Références:

Ref $1$: Henriette Elvang, Yu-tin Huang: Scattering Amplitudes

Ref $2$: Jaroslav Trnka: The Amplituhedron

[For simplicity, the notations of the $2$ refs have been merged]

The area of a triangle in $CP_2$, can be expressed, using dual space complex 3-dimensional quantities $Z_i$, as (Ref $1$, page $157$, formula $10.17$):

$$A = frac{1}{2} frac{langle123rangle^2}{langle012ranglelangle023ranglelangle031rangle}$$
where $i$ is for $Z_i$, $Z_0 = ^t(0,0,1)$, and $langle abc rangle=det(abc).$

On the other way, there is a “canonical form” (Ref $2$, page $26$):

$$Omega_p= frac{langle Y dY dYrangle langle123rangle^2}{langle Y12ranglelangle Y23ranglelangle Y31rangle},$$ where $Y$ represents a point in the interior of the triangle.

The relation between the $2$, (see Ref 2, pages $31,32$), seems to be an integration: $A =int delta(Y -Z_0) Omega_p$

If the elements above are correct, that I don’t understand is the utility of the canonical form, because in the integration, we keep only one point $Y = Z_0$, so the “integration” is somewhat “trivial”, so why is used this presentation with the canonical form (which is linked to the grassmannian)?