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I How to Define a Shape

  1. Mar 28, 2016 #1
    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.
     
  2. jcsd
  3. Mar 28, 2016 #2

    andrewkirk

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    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.
     
  4. Mar 28, 2016 #3
    Ah okay, that makes sense.
    Now I'm curious, what inequalities are needed to describe curvilinear shapes?
     
  5. Mar 28, 2016 #4

    andrewkirk

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    The most famous one is ##x^2+y^2\leq 1##
     
  6. Mar 29, 2016 #5

    Mark44

    Staff: Mentor

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