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The circle is one of the first shapes we learn as children. It has been known since before the beginning of recorded history. We, like ancient peoples, can observe natural circles all around us in the moon and the sun, in flora and fauna. The circle is responsible for the development of the wheel, gears, geometry, mathematics, astronomy, all of which arguably made modern civilization possible.
The word "circle" derives from the Greek, "kirkos", based on the word ker - "to turn or bend". The term "Monad", meaning "unit" was used by Greek philosophers from Pythagoreans to Plato, Aristotle, and Plotinus in reference to a variety of entities from a circle, to a genus, or even God. Most medieval scholars, as concerned with divinity as geometry, also believed the perfect circle contained something intrinsically divine.
The following article demonstratest the John W. Clark method of creating an accurate circle within a square. This method can also be applied in perspective to create an ellispse.
To divide the square into four equal quadrants, draw diagonals AB, and CD. Connect line EF through the intersection of AB/CD. Draw line GH through the intersection of AB/CD.
Square now divided into four equal Quadrants
If a square is divided into four equal Quadrants, note that circle touches the square at points A, B, C and D. One would think that drawing an arc for the circle with these points within each Quadrant would produce the desired circle. However, having one additional point of reference within each Quadrant would be very useful.
In Quadrant I, find the center by drawing lines 1,2 and 3,4. Where they intersect, draw line 5,6 which divides Quadrant I into equal horizontal parts.
Draw lines AB and CB
Finally, draw line xz.
Where line xz intersects line AB, the additional point that allows us to draw the arch of the circle in Quadrant I is established.
This shows where this point will fall on the circle.
This shows where this point will fall on the circle in Quadrant I.
Flip the lines in Quadrant I to the lower Quadrant III.
Repeat for the lower Quadrant IV.
Key points shown in all Quadrants.
Key points shown for the drawing of the circle within the square.
It also works in perspective.
Construction of a circle on a curved surface (upper portion).
Construction of a circle on a n any curved surface (lower portion).
Circle drawn on a curved surface.
Proof that a circle on a curved surface looks distorted.
Another example of circles on curved surfaces (wings and fuselage of the airplanes).
Ellipses:
An ellipse is a circle seen on edge. The circle in the center of this illustration is rotated in both the left and right images. This produces an ellipse.
Ellipse slices through a full circle.
Ellipse slices through a full circle.
Ellipse slices through a full circle.
Ellipse slices through a full circle.
