Drawing showing the situation described below

As the title says, i want to project 3D points with known (x, y, z) coordinates into a 2D plane with (x', y') coordinates, knowing that the x and y axes are respectively identical to the x' and y' axes ( The (OXY) plane is the same as the (OX'Y') plane) and they have the same measure unit.

So basically what i need is the formula to transform the a point's (x, y, z) coordinates into (x', y') coordinates with a perspective projection ( The further an object is from the 2D plane the smaller it seems).

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    $\begingroup$ Your diagram shows "orthogonal projection", $(x, y, z) \mapsto (x, y)$. Is that what you mean by "perspective projection"? If not, could you please say a bit more about what perspective projection means? $\endgroup$ Jun 1, 2017 at 15:59
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    $\begingroup$ @AndrewD.Hwang When googling the question i found it could be done with just discarding the z component , it was called the orthogonal projection , and that's not what i want , i want the depth to be seen ( the more an object is far from the 2D plane the smaller it seems ), so that's why i said a perspective projection $\endgroup$
    – Put Me
    Jun 1, 2017 at 16:10
  • $\begingroup$ One common approach is to put a "light source" at $(0, 0, h)$, i.e., along the $z$-axis at distance $h$, and to let each point cast a "shadow" in the $(x, y)$-plane. A common formula for this is$$(x, y, z) \mapsto \frac{h}{h - z}(x, y),$$which works as expected so long as $z < h$, i.e., points do not "go above" the light source. (As $h \to \infty$, projection looks more and more like orthogonal projection.) Does that do what you want? $\endgroup$ Jun 1, 2017 at 16:17
  • $\begingroup$ I missed your edit in making the preceding comment. If you know $z > 0$ for all your points, the formula $$(x, y, z) \mapsto \frac{h}{h + z}(x, y)$$has the property you requested (objects further from the $(x, y)$-plane "cast a smaller shadow"). $\endgroup$ Jun 1, 2017 at 16:21
  • $\begingroup$ z could go underneath 0 , as object will be moving . As i'm learning to program 3D games,obviously objects will be moving . $\endgroup$
    – Put Me
    Jun 1, 2017 at 16:28

1 Answer 1


3D projection is really very simple. The hard part is understanding how it is done; and that is what I shall try to explain here. It is all based on optics, and (linear) algebra.

Let's assume you stand in front of a window, looking out. If you stand in the center of the window, looking out through the center of the window, then we can treat the center of your eye (more precisely, the center of the lens in the pupil of your dominant eye) the origin in 3D coordinates. Using OP's conventions, $x$ axis increases up, $y$ axis right, and $z$ axis outside the window. Thus, the center of the window is at $(0, 0, d)$, where $d$ is the distance from the eye to the window.

If we know the 3D coordinates in the above coordinate system of interesting details outside, 3D projection tells us their coordinates on the surface of the window. These coordinates are what OP needs to draw 3D pictures to a 2D surface.

Here is a rough diagram of the situation: Perspective projection diagram The blue pane is the window, the eye is at the lower left corner, and we are interested in the projected coordinates (projected to the window, that is) of the four corners of some cube at some distance.

In a very real sense, those coordinates are obtained by linear interpolation, except that one end of the line segment is at the eye (which we already decided is the origin, so coordinates $(0, 0, 0)$, the other end is at the 3D coordinates of the detail we wish to project, and the interpolation point is where that sight line (usually called "ray") intersects the view plane (the window, in our case).

Let's say one of the 3D coordinates of an interesting detail, say a corner of the greenish cube above, are $(x , y , z)$. That ray intersects the window at $$\begin{cases} x' = x \frac{d}{z} \\ y' = y \frac{d}{z} \\ z' = z \frac{d}{z} = d \end{cases}$$ Therefore, the 2D coordinates of that detail on the window are $$(x' , y') = \left( x \frac{d}{z} ,\, y \frac{d}{z} \right) = \frac{d}{z} ( x , y )$$ or, in other words, you simply multiply the $x$ and $y$ 3D coordinates by $d/z$.

If some detail has a $z$ coordinate smaller than $d$, it is basically between the window and the eye. These are problematic, because projecting them to the window no longer makes sense wrt. optics. In visualization software, such details are normally simply not drawn; then, you can think of the window as being the "camera surface" of some pyramid-shaped probe, the tip of the pyramid being the 3D coordinate system origin.

In games, details with $z$ coordinate smaller than $d$ often produce visual glitches, like seeing through a wall, or similar. The real numerical problems occur near the origin, near the eye. If any graphical primitive, like a line, plane, or sphere, intersects with the eye, it means the eye is badly hurt in real life; with 3D projection, we get unrealistic results (like polygons getting twisted through origin), because our model breaks down at the eye point.

I very warmly recommend you familiarize yourself with basic 2D and 3D vector algebra. Using vector addition, subtraction, scaling, dot product, and cross product, many of the operations you need to work with 3D worlds become much simpler.

Do not bother with Euler angles (or Tait-Bryan angles); they have limitations (especially gimbal lock) and ambiguities (the order of the rotations). Instead, learn about versors, or unit quaternions that represent rotations. They have four components, $$\mathbf{q} = ( w , i , j , k )$$ where $w^2 + i^2 + j^2 + k^2 = 1$. You can easily interpolate ("blend") between different versors, for example to simulate a camera panning and rotating from one orientation to the next.

Although quaternions have a reputation (among programmers) of being hard to grok, their unit quaternion or versor subset is actually very programmer-friendly. They are numerically stable (you can always divide the components by $\sqrt{w^2 + i^2 + j^2 + k^2}$ to scale it back to unit length, and it won't bias the rotation in any specific way).

For computation, you expand (convert) the versor to a 3×3 rotation matrix, which unlike those for Euler or Tait-Bryan angles, is unique for versors. There are no "gotchas" or ambiguities.

You will also need to learn about matrix-matrix multiplication, and matrix-vector multiplication, so that you can efficiently apply the rotations to vectors.

Matrix-matrix multiplication is used to combine rotations or transformations described by matrices, to other such matrices. This means you only need to use one matrix to transform any vector, but that matrix can be the result of several different transformations itself. (For example, if you have a robot arm with ball joints, you can describe each joint using a versor, and a rotation matrix derived from that versor. When you start at the base, you simply multiply the current transformation matrix by the ball joint transformation matrix, to get the local coordinate system in the part that follows each ball joint.)

Versor-versor multiplication (Hamilton product) does the exact same for versors: multiplying $\mathbf{q}_1 \mathbf{q}_2$ yields a versor that represents a rotation by versor $\mathbf{q}_1$ followed by a rotation by versor $\mathbf{q}_2$. (Numerically, you'll want to divide each component of the result by $\sqrt{w^2 + i^2 + j^2 + k^2}$, to ensure it has unit length; as I wrote earlier, this impacts no bias to the result, and allows you to apply as many consecutive rotations as you want, without any distortion -- unlike for example for matrices, which would require orthonormalization after a dozen or so steps, even if using double-precision floating-point numbers.)

The reverse problem, trying to find the object and the point on an object, when you know the ray arriving at the eye, is called ray casting. If you then continue to trace the possible rays, to find out which ones might originate in light sources, you get to ray tracing.

If you transform the 3D coordinate system so that your eye (or camera) is always at origin, the intersection tests become much easier. In particular, consider the sphere case: let $\vec{c}$ is the center of the sphere, $r$ is its radius, and $\hat{n}$ is the unit vector ($\lVert\hat{n}\rVert = 1$) showing the direction where the ray came to the eye. Let $$D = r^2 + \left ( \hat{n} \cdot \vec{c} \right )^2 - \vec{c} \cdot \vec{c}$$ If $D \lt 0$, the ray did not intersect the sphere. If $D = 0$, the ray grazed the sphere, i.e. intersected it tangentially (at one point). If $D \gt 0$, the ray intersects the sphere at distance $R$ from origin: $$R = \hat{n} \cdot \vec{c} \pm \sqrt{ D }$$ at point $R \hat{n}$. If we are outside the sphere, use $-$ above; if we are inside the sphere, use $+$ above. In general, pick the sign that yields the smaller, but positive, $R$.

Since ball-and-stick models are often a favourite starting point for physicists and chemists interested in 3D graphics, I've collected the formulas needed to do the above with cylinders (without end caps, with flat end caps, or with spherical end caps) to my Wikipedia user page.

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    $\begingroup$ I wish I could favorite an answer $\endgroup$ Aug 23, 2017 at 12:58
  • $\begingroup$ This answer is absolutely amazing! The only followup question I have is whether you've ever found a good reference for chaining transformations when you have a quaternion to signify the rotation and an associated translation. The simple approach is to convert the quaternion into a rotation matrix, and then create an affine transformation matrix with the translation, but I've found the math tricky if I don't want to do the back and forth between the two representations. $\endgroup$
    – Olshansky
    Feb 17, 2019 at 1:40
  • $\begingroup$ I'm really just trying 3D projection but has a glitch you can see in this video: youtu.be/XXqI-w_iaSE So when the grid gets behind the eye, it gets inverted. If i'm not adding ex,ey to the linear interpolation formula (wiki 3d projection page - display surface) then the glitch does not appear. $\endgroup$ Dec 29, 2019 at 23:18

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