I How to Define a Shape Based on Variables

[Berenices]

Hello all,
I am not too experienced with geometry. I am just curious whether it would be possible to define a shape based on variables.
Say you have a simple relationship between volume and some variables. V=x+y. This tells you about the volume of a 3D object, however, it does not describe the shape of the object in question. How would you write a relationship that describes both a volume and a shape?
Thanks in advance.

 

[andrewkirk]

Consider a triangle on the number plane. If we are talking about the inside of the triangle together with its boundary then it is defined by three inequalities using a coordinate system. For instance the following defines the shape that is the triangle with corners at (0,0), (1,0) and (0,1)

$$(x\geq 0)\wedge (y\geq 0)\wedge (x+y\leq 1)$$

where $\wedge$ means 'and'.
This is the intersection of three half-planes, bordered by the lines that, segments of which make up the three sides of the triangle.

We can take exactly the same approach on a general manifold in diff geom. We can define the n-dimensional equivalent of a n-polygon in an n-dimensional manifold as:

$$\left(\sum_{k=1}^n a_{1k}\leq b_1\right)\wedge ... \wedge \left(\sum_{k=1}^n a_{nk}\leq b_n\right)$$

This linear approach only works for linear-bounded shapes. Other inequalities are needed for curvilinear shapes, just as we use a different equation in 2D to define a circle.

 

[Berenices]

Ah okay, that makes sense.
Now I'm curious, what inequalities are needed to describe curvilinear shapes?

 

[andrewkirk]

The most famous one is $x^2+y^2\leq 1$

 

[Mark44]

Thread moved, as this question has nothing to do with differential geometry.