First fundamental form

The aboriginal axiological anatomy is a boxlike form

on the departure even to the apparent which is acclimated to account distances and angles. For a parametrized apparent its coefficients can be computed as follows:

Arc breadth of parametrised curves on the apparent S, the bend amid curves on S, and the apparent breadth all accept expressions in agreement of the aboriginal axiological form.

If (u(t), v(t)), a ≤ t ≤ b represents a parametrised ambit on this apparent again its arc breadth can be affected as the integral:

The aboriginal axiological anatomy may be beheld as a ancestors of absolute audible symmetric bilinear forms on the departure even at anniversary point of the apparent depending calmly on the point. This bend helps one account the bend amid two curves on S intersecting at a accustomed point. This bend is according to the bend amid the departure vectors to the curves. The aboriginal axiological anatomy evaluated on this brace of vectors is their dot product, and the bend can be begin from the accepted formula

expressing the cosine of the bend via the dot product.

Surface breadth can be bidding in agreement of the aboriginal axiological anatomy as follows:

The announcement beneath the aboveboard basis is absolutely , and so it is carefully absolute at the approved points.