Assuming we can construct parabola, hyperbola geometrically just as circle with compass and ellipses with string. What new things can be constructed can constructed. For, example we can double the cube by using parabola and circle i.e. we can construct cube-root of 2.

For, example Can we do things like trisecting an angle or dividing them into n parts, or construction of heptagon?

EDIT: Thanks for reply. I want to give context for my questions. As a personal project, I want to make a puzzle game or app. The UI allows user to create that any conics that are circle, parabola, ellipse, hyperbola, ruled line. There are various parameters that can used to create these conics. In each puzzle some constructions are needed to solve the problems. I need various geometric results and such. That is the gist of it. Therefore, while gather information around this topics, I also need do devise some of my own results and objectives, to make various problems. I can use some guidance.

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    $\begingroup$ Welcome to MSE! Can you be more precise with your query? The problem can have multiple answers! $\endgroup$
    – Anand
    Aug 27 '20 at 12:03
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    $\begingroup$ Two conics over $K$ intersect in a point with coordinates of degree at most $4$ over $K$ and I think you can probably get any degree $4$ or smaller over $K$ integer in this way, and in general towers of fields of that form $\endgroup$
    – user208649
    Aug 27 '20 at 13:19
  • $\begingroup$ I give 20 references about conic constructible numbers in this 17 December 2008 sci.math post. See also "Some Remarks on Conic Constructible Numbers" on pp. 6-7 of this manuscript. $\endgroup$ Aug 28 '20 at 5:59
  • $\begingroup$ Off the top of my head, angle trisection and cube roots become possible rather immediately. I dont know what else falls out, but it solves some age old problems. Not squaring the circle - thats still impossible (because pi is transcendental). But the heptagon becomes constructible. $\endgroup$ Sep 6 '20 at 16:30

Check out N. Sinclair's Mathematical Applications of Conic Sections in Problem Solving in Ancient Greece and Medieval Islam, which discusses how the geometers of antiquity used conics for constructions such as doubling of the cube and angle trisection.

The ancient Greeks had a special classification scheme for geometrical problems. Pappus, who flourished at the beginning of the 4th century A.D., remarks in his Collection that the ancients divided problems into three classes: 'plane', 'solid', and 'curvilinear'. 'Plane' problems could be solved by means of ruler and compass; 'solid', by means of one or more sections of the cone but not by 'plane' methods; 'curvilinear', by means of special curves, but not by 'plane' or 'solid' methods. He notes that both the cube duplication and the angle trisection fall within the 'solid' class, and that this posed problems for researchers, who were not able to construct conics in the plane.

I personally ran into this topic when I was studying certain constructions concerning two non-intersecting conics.


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