Derivation of Σ GL perspective projection matrix

Computer monitors are two-dimensional surfaces . from OpenGL Rendered 3D The scene must be as 2D The image is projected onto the computer screen . The projection matrix is used for this projection transformation . First , It converts all vertex data from eye coordinates to clipping coordinates . then , By matching with the clipping coordinates w Component division , These clipping coordinates are also converted to normalized device coordinates (NDC).

Clipping coordinates ： The eye coordinates are now multiplied by the projection matrix , Become clipping coordinates . The projection matrix defines the visual cone —— How vertex data is projected onto the screen （ Perspective or orthogonal ）. It's called clipping coordinates , Because the transformed vertices (x,y,z) It's through talking to ±w_clip Cut by comparison .

Cone culling ( tailoring ) The operation is performed in clipping coordinates , Just divided by w_clip Before . Through and with w_clip Comparison , The clipping coordinates are tested x_clip、y_clip and z_clip. Because divided by w_clip Then become normalized NDC coordinate , So to satisfy -1<=x_clip/w_clip<=1, therefore x_clip∈[-w_clip,w_clip]; Empathy y_clip∈[-w_clip,w_clip],z_clip∈[-w_clip,w_clip]. If any clipping coordinates are less than -w_clip Or greater than w_clip, The vertex will be discarded ( Cut out ).

therefore , We must remember , tailoring （ Cone culling ） and NDC The transformations are integrated into the projection matrix . Here's how to start from 6 Construct the projection matrix with two parameters ：left、right、bottom、top、near and far The boundary value .

then ,OpenGL The edges of the clipped polygon will be reconstructed ( See the two red lines in the figure below ).
The gray area in the following figure is the point that is reserved but not discarded , Satisfy ：x_c,y_c,z_c∈(-w_c,w_c)

The following figure shows the perspective cone and normalized equipment coordinates (NDC)：

In perspective projection , The cone of the visual cone （ Eye coordinates ） Medium 3D Points are mapped to cubes (NDC);x The coordinates are from [l,r] To [-1,1],y The coordinates are from [b,t] To [-1,1],z The coordinates are from [-n,-f] To [-1,1].

void glFrustum(
// Specify the coordinates of the left and right vertical clipping planes .
GLdouble left, GLdouble right,
// Specify the coordinates of the bottom and top horizontal clipping planes .
GLdouble bottom, GLdouble top,
// Specifies the distance to the near depth clipping plane and the far depth clipping plane . Both distances must be positive .
GLdouble nearVal, GLdouble farVal
);

Please note that , Eye coordinates are defined in the right-hand coordinate system , but NDC Use the left-hand coordinate system . in other words , The camera at the origin moves along in eye space -Z Axis view , But in NDC Middle edge +Z Axis view . because glFrustum() We only accept near and far Positive value of , Therefore, we need to negate them when constructing projection matrices .

stay OpenGL in , One in eye space 3D The point is projected to the near plane ( The projection plane ) On . The following figure shows a point in eye space (x_e,y_e,z_e) How to project to the near plane (x_p,y_p,z_p).

From the of the visual cone [ Top view ], That is, the of eye space x coordinate ,x_e Mapped to x_p,x_p It is calculated by using the ratio of similar triangles ：

From the side of the visual cone ,y_p Calculated in a similar way ：

Be careful x_p and y_p Rely on a z_e; They are associated with -z_e In inverse proportion . let me put it another way , They are all -z_e except . This is the first clue to construct the projection matrix . After transforming the eye coordinates by multiplying the projection matrix , The clipping coordinates are still homogeneous . It eventually becomes standardized equipment coordinates (NDC), Divided by the clipping coordinates w component .（ For more details , see OpenGL_Transformation)

therefore , We can cut the coordinates of w The component is set to -z_e. The second of the projection matrix 4 Line into (0,0,-1,0).

Next , We use a linear relationship to describe x_p and y_p Mapping to NDC Of x_n and y_n;[l,r]⇒ [-1,1] and [b,t]⇒ [-1,1].

then , We will x_p and y_p Substitute into the above equation .

Be careful , For perspective Division (x_c/w_c,y_c/w_c）, Let's make two terms of each equation be -z_e to be divisible by . We will w_c Set to -z_e, The terms in parentheses become the of clipping coordinates x_c and y_c.

From these equations , We can find the second of the projection matrix 1 Xing He 2 That's ok .

Now? , We just need to solve the problem 3 Row projection matrix . Find out z_n A little different from others , Because of... In eye space z_e Always projected onto the near plane -n. But we need the only z Values for clipping and depth testing . Besides , We should be able to cancel the projection （ inverse transformation ）. Because we know z Don't depend on x or y value , So we borrow w Component to find z_n and z_e The relationship between . therefore , We can specify the second of the projection matrix in this way 3 That's ok .

In eye space ,w_e be equal to 1. therefore , The equation becomes ：

To find the coefficient A and B, We use (z_e,z_n) Relationship ：(-n,-1) and (-f,1), And put them into the above equation , figure out A、B.

here ,z_e And z_n The relationship becomes ：

Last , We found all the entries of the projection matrix . The complete projection matrix is ：

The projection matrix is suitable for general viewing cone . If the visual body is symmetrical , namely r=-l, t=-b, It can be reduced to ：

Before we move on , Please look again z_e and z_n The relationship between , equation （3）. You notice that this is a rational function ,z_e and z_n There is a nonlinear relationship between （ Here's the picture ）. This means that the accuracy in the near plane is very high , But the accuracy in the far plane is very low . If the range [-n,-f] More and more big , It will lead to the problem of depth accuracy （z-fighting）; Near the far plane z_e Small changes in do not affect z_n value .n and f The distance between should be as short as possible , To minimize the depth buffer accuracy problem .

Use FOV Specified perspective projection ：


h = 2 × near × tan(θ/2)
w = h × aspect;
Corresponding to the above perspective projection matrix ：
r = w/2;t = h/2
therefore ：
n/r = (2×n)/w
= (2×n)/(h × aspect)
= (2×n)/(2 × n × tan(θ/2) × aspect)
= cot(θ/2)/aspect
Similarly obtained ,n/t = cot(θ/2)
Then the perspective projection matrix is ：