The Ellipse You See in Perspective and the Ellipse You Draw Are Not the Same
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I saw someone on X claiming something like “a circle looks like the same ellipse from any perspective.”
I couldn’t trace the original post, and it was probably simplified for a drawing context.
But mathematically, at least three separate problems are tangled up here:
- What happens to a circle in 3D space when viewed through a camera?
- Is an ellipse drawn directly on paper the same as the ellipse produced by projecting a circle?
- How do the major axis, minor axis, and center of the ellipse correspond to the original circle’s center and diameter?
In the article where I tested camera angles with AI image generation, angle specifications worked surprisingly loosely.
This time the topic is not image generation but the geometry underneath.
”A circle viewed at an angle becomes an ellipse” is roughly correct, but “so you can just treat it as a normal ellipse” jumps too far.
Parallel Projection Keeps Things Simple
Set aside perspective for a moment.
Imagine a camera infinitely far away, projecting objects onto paper with parallel rays.
This is parallel projection, or more generally an affine transformation.
In this case, a circle maps to an ellipse.
The center maps to the center.
The midpoint of a diameter stays a midpoint after projection.
In matrix form, given a circle in 2D
we are simply applying a linear transformation on the plane
If compresses the vertical direction, the circle becomes an ellipse.
Within this framework, “a circle viewed at an angle is an ellipse” is a pretty clean explanation.
The drawing instruction of “fit the ellipse inside a box” or “draw the cross-section of a cylinder as an ellipse” mostly relies on this parallel-projection intuition.
For isometric views or weak perspective, it works well enough.
Perspective Projection Breaks Centers and Midpoints
The trouble starts with perspective projection.
In a pinhole camera model or one-point/two-point perspective, closer objects appear larger and farther objects appear smaller.
This is not an affine transformation but a projective transformation.
Under a projective transformation, a circle still maps to one of the conics: an ellipse, parabola, or hyperbola.
For a normal camera viewing a finite disc, the outline on the image is usually an ellipse.
However, projective transformations do not preserve midpoints.
They do not preserve distances or angles either.
In general, “the projection of the circle’s center” and “the center of the ellipse on the image” do not coincide.
This is where drawing-related confusion starts.
Why the offset?
In perspective, the far half of the circle shrinks relative to the near half.
Even though the distance from the center to the near edge and the far edge is the same on the original circle, they don’t appear equal on the image.
So the projected center of the circle doesn’t land at the midpoint of the ellipse.
If you ignore this and place something on the major axis of the on-screen ellipse, you’ll think you’re targeting the center of the disc in space, but it will look off.
The complaint “when I actually put things on this line and view at an angle, they don’t look centered” is exactly this phenomenon.
The Major Axis Is Just the Longest Line on the Image
As a 2D figure, an ellipse has a clear major axis.
It’s the longest chord passing through the center.
For a horizontally oriented ellipse, that’s the horizontal direction.
But that’s a fact about “the ellipse as a 2D shape on the image.”
It’s separate from the 3D question of which diameter of the original circle mapped there, or where the circle’s center ended up on that line.
What perspective projection strongly preserves is limited:
| Preserved | Not preserved |
|---|---|
| Points map to points | Distances |
| Lines map to lines | Angles |
| Incidence (point on a line) | Midpoints |
| Intersections | Circle centers |
A “diameter of the circle” does map to some line segment after projection.
But the midpoint of that segment won’t necessarily be the center of the ellipse on the image.
Conversely, if you back-project the major axis of the on-screen ellipse, it may correspond to a chord that doesn’t pass through the center of the original disc.
This is why “the longest line of the ellipse is obviously the horizontal axis” and “but things placed there don’t look centered” can both be true at the same time.
The first is a statement about the 2D ellipse’s metric.
The second is about projecting the 3D disc’s center.
The word “center” refers to two different things.
The Ellipse as a Projected Circle vs. Just an Ellipse
Draw an ellipse freehand on paper and it’s of course an ellipse.
But an ellipse that represents “a specific circle viewed from a specific camera position” carries additional constraints.
If you’re drawing the opening of a cylinder in one-point perspective, it’s not just the ellipse in isolation — the plane the circle sits on, the vanishing point, the bounding quadrilateral, and the depth direction all come as a package.
Extracting just the ellipse and saying “the major axis is here” won’t recover the original disc’s center or the cylinder’s axis.
The drawing technique of “inscribe a circle inside a perspective quadrilateral” exists for this reason.
Instead of drawing the circle as a standalone ellipse, you first project the square containing the circle.
Then you use the tangent points on the quadrilateral’s edges and its diagonals to carry the spatial relationships.
But even here, caution is needed.
The diagonal intersection of the projected square is the image of the original square’s center.
That point is not necessarily the center of the ellipse on the image.
When drawing center lines, mixing up “the image of the disc’s center” with “the geometric center of the ellipse as a 2D shape” leads to errors.
In the Formula, the Denominator Is the Culprit
In the simplest form of perspective projection, a 3D point maps to an image point as:
is the focal length, is the depth from the camera.
In parallel projection, it would be something like , — no denominator.
In perspective, dividing by makes objects farther away appear smaller.
When a disc is tilted relative to the camera, each point on the circumference has a slightly different .
Near-side points are divided by a smaller , far-side points by a larger .
This difference in denominators creates the center offset.
In projective geometry terms, perspective projection can be written as a linear transformation in homogeneous coordinates:
The final division by is the key.
Because of this division, midpoints and lengths are not preserved.
On the other hand, lines and intersections are preserved, which is why vanishing points, construction lines, and tangent lines are the powerful tools in perspective drawing.
As with the article on derivatives where I diagrammed contour lines and gradients, looking at “what is preserved” cuts through the confusion in geometry problems.
In perspective projection, the projective-geometry invariants of lines, intersections, and tangencies are more fundamental than the Euclidean ones of distance, angle, and center.
Wide-Angle vs. Telephoto Change the Projection’s Behavior
Everything above assumed a pinhole camera.
In real photography, changing the focal length changes how the same disc looks.
A wide-angle lens has a broad field of view and is typically used close to the subject.
Being close means the relative difference in between the near and far sides of the disc is large.
The denominator in varies significantly across points, so the offset between the ellipse center and the projected circle center is noticeable.
Faces getting stretched at the edges of a group photo is this projective distortion in action.
Telephoto lenses work the opposite way: they crop from a distance.
Each point on the subject has roughly the same , so the denominator is nearly constant.
The behavior approximates parallel projection, and the center offset is negligible.
The compression effect where telephoto shots flatten depth is exactly this.
Bringing it back to drawing: the “center of a circle drawn as an ellipse looks off” problem is more pronounced the wider the field of view, and barely noticeable with a telephoto-like viewpoint.
Lens Distortion and Smartphone AI Correction
One more factor: real lenses have optical distortion.
In an ideal perspective projection, straight lines map to straight lines, but physical lenses bend lines near the edges of the frame.
Wide-angle lenses commonly exhibit barrel distortion, where the edges bulge outward.
Telephoto lenses tend toward pincushion distortion, where edges are pulled inward.
With distortion present, a circle in space doesn’t even map to a proper ellipse on the image.
A nonlinear warp sits on top of the projective transformation, making the outline slightly egg-shaped.
Lens profile correction in cameras and RAW processing software reverses this optical distortion to recover the ideal perspective projection.
After correction, straight lines are straight again, and the image of a circle becomes the ellipse that projective geometry predicts.
Lens correction doesn’t break the projective transformation — it brings the image closer to one.
Modern smartphones go a step further.
When taking a selfie with a wide-angle front camera, faces get stretched at the edges — and that’s the mathematically correct result of perspective projection.
The “face looks distorted” effect is not an optical error but the mathematical consequence of wide angle plus close range.
Yet iPhone and Pixel camera apps detect face regions and correct the projective distortion.
It’s essentially pushing the perspective projection partway back toward parallel projection, breaking a geometrically correct projection for the sake of a “natural-looking” result.