Functions homo/isomorphic to change in scale

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SUMMARY

The discussion focuses on functions that maintain their structural properties when scaled, specifically within the context of projections. It establishes that conic sections, such as parabolas and circles, exhibit distinct behaviors under projection, with parabolas retaining their form while circles do not. The conversation highlights the importance of projective surfaces and homogeneous coordinates in understanding these properties. The assertion that all conic sections share this scaling property is debated, emphasizing the need for clarity in definitions.

PREREQUISITES
  • Understanding of conic sections, including parabolas and circles.
  • Familiarity with projective geometry and projective surfaces.
  • Knowledge of homogeneous coordinates and their applications.
  • Basic principles of mathematical projections and transformations.
NEXT STEPS
  • Research the properties of conic sections in projective geometry.
  • Explore the concept of homogeneous coordinates and their role in projections.
  • Study the differences between various types of projections in mathematics.
  • Investigate the implications of scaling transformations on different mathematical functions.
USEFUL FOR

Mathematicians, students of geometry, and anyone interested in the properties of functions under scaling and projection transformations.

LogicalTime
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I would like to find out which functions retain the same structure when they are scaled. Particularly I am interested in projections.

For example, a parabola 3d space viewed at another angle can still be represented by at^2 + bt+c. A circle however can not (ellipse)

I am guessing conic sections have this property? Are there any other functions that have this property, and what terms are associated with this property?
 

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I think it would help looking at projective surfaces, with homogeneous coordinates.
 
LogicalTime said:
I would like to find out which functions retain the same structure when they are scaled. Particularly I am interested in projections.

For example, a parabola 3d space viewed at another angle can still be represented by at^2 + bt+c. A circle however can not (ellipse)

I am guessing conic sections have this property? Are there any other functions that have this property, and what terms are associated with this property?
Have what property? You assert that a parabola, projected onto any plane, is still a parabola (which is not quite true- it can project to a ray) while that is not true for a circle. But parabola and circle are both conic sections.
 

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