Proving Ellipse When Viewing Circle with Non-Perpendicular Line of Sight

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SUMMARY

The discussion centers on the geometric transformation of a circle into an ellipse when viewed from a non-perpendicular line of sight. It establishes that this transformation can be proven using the definition of an ellipse as a conic section, specifically through the intersection of a plane and a circular cone. The use of Dandelin spheres is highlighted as a method for proving this relationship. Additionally, the conversation addresses common misconceptions about the appearance of the ellipse, clarifying that it does not resemble a chicken egg shape when viewed obliquely.

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  • Understanding of conic sections and their properties
  • Familiarity with geometric transformations, specifically affine and projective transformations
  • Knowledge of Dandelin spheres and their role in geometry
  • Basic principles of perspective projection and parallel projection
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Draw a circle in a paper, if the line of sight perpendular to the paper , we see a circle ,but if the line of sight is not perpendular to the paper, must we see an ellipse ? how to prove it ? how to find out the major axis or minor axis ? It seem that when we observe the circle ,it should be wide in near place and narrow in far place and should look like a chicken egg ,but it is not true ,how to explain the contradiction. When we view by eyes,we make an affine transformation or a projective transformation ? Whether the relation is true below : parallel projection = affine transformation, central projection = Perspective projection?
 
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redoxes said:
Draw a circle in a paper, if the line of sight perpendular to the paper , we see a circle ,but if the line of sight is not perpendular to the paper, must we see an ellipse ? how to prove it ? how to find out the major axis or minor axis ? It seem that when we observe the circle ,it should be wide in near place and narrow in far place and should look like a chicken egg ,but it is not true ,how to explain the contradiction.

Hi redoxes! Welcome to PF! :smile:

Proof that a circle viewed obliquely is an ellipse depends on how you define an ellipse.

The easiest way is to define it as a conic section, that is as the intersection of a plane and an upright circular cone …

then you can prove it using Dandelin spheres …

see the PF LIbrary on https://www.physicsforums.com/library.php?do=view_item&itemid=98" :wink:
 
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