Why does my perspective implementation fails in di

问题:

I wrote a program that takes in entry some points, expressed in 3D coordinates and that must be drawn in a 2D canvas. I use perspective projection, homogeneous coordinates and similar triangles to do that. However, my program does not work and I actually don't know why.

I followed two tutorials. I really understood the geometrical definitions and properties I have read. However, my implementation fails... I will write references to these both courses little by little, to make your reading more confortable :).

Overview : geometrical reminders

The perspective projection is done following this workflow (cf. these 2 courses - I wrote pertinent links about the latter (HTML anchors) further down, in this post) :

1. Definition of the points to draw, expressed according to the world's coordinates system ; Definition of the matrix of projection, which is a matrix of transformation that "converts" a point expressed according to the world coordinates system into a point expressed according to the camera's coordinates system (NB : I believe this matrix can be understood as being the 3D object "camera")

2. Product of these points with this matrix (as defined in the adequat part, below, in this document) : the product of these world-expressed points results in the conversion of these points to the camera's coordinates system. Note that points and matrix are expressed in 4D (concept of homogenous coordinates).

3. Use of similar triangles concept to project (only computing is done at this step) on the canvas the in-camera-expressed points (using their 4D coordinates). After this operation, the points are now expressed in 3D (the third coordinate is computed but not actually used on the canvas). The 4th coordinate is removed because not useful. Note that the 3rd coordinate won't be useful, except to handle z-fighting (though, I don't want to do that).

4. Last step : rasterization, to actually draw the pixels on the canvas (other computing AND displaying are done at this step).

First, the problem

Well, I want to draw a cube but the perspective doesn't work. The projected points seem to be drawn withtout perspective.

What result I should expect for

The result I'm expecting is the cube displayed in "Image" part of this below PNG :

What I'm outputting

The faces of my cube are odd, as if perspective wasn't well used.

I guess I know why I'm having this problem...

I think my projection matrix (i.e. : the camera) doesn't have the good coefficients. I'm using a very simple projection matrix, without the concepts of fov, near and far clipping planes (as you can see belower).

Indeed, to get the expected result (as previouslyt defined), the camera should be placed, if I'm not mistaken, at the center (on axes x and y) of the cube expressed according to the world coordinate system and at the center (on axes x and y) of the canvas, which is (I make this assumption) placed 1 z in front of the camera.

The Scastie (snippet)

NB : since X11 is not activated on Scastie, the window I want to create won't be shown.

https://scastie.scala-lang.org/N95TE2nHTgSlqCxRHwYnxA

Entries

Perhaps the problem is bound to the entries ? Well, I give you them.

Cube's points

Ref. : myself

val world_cube_points : Seq[Seq[Double]] = Seq(
  Seq(100, 300, -4, 1), // top left
  Seq(100, 300, -1, 1), // top left z+1
  Seq(100, 0, -4, 1), // bottom left
  Seq(100, 0, -1, 1), // bottom left z+1
  Seq(400, 300, -4, 1), // top right
  Seq(400, 300, -1, 1), // top right z+1
  Seq(400, 0, -4, 1), // bottom right
  Seq(400, 0, -1, 1) // bottom right z+1
)

Transformation (Projection) matrix

Ref. : https://www.scratchapixel.com/lessons/3d-basic-rendering/perspective-and-orthographic-projection-matrix/building-basic-perspective-projection-matrix , End of the Part. "A Simple Perspective Matrix"

Note that I'm using the simplest perspective projection matrix : I don't use concept of fov, near and far clipping planes.

new Matrix(Seq(
  Seq(1, 0, 0, 0),
  Seq(0, 1, 0, 0),
  Seq(0, 0, -1, 0),
  Seq(0, 0, -1, 0)
))

Consequence of this matrix : each point P(x;y;z;w) producted with this matrix will be : P'(x;y;-z;-z).

Second, the first operation my program does : a simple product of a point with a matrix.

Ref. : https://github.com/ssloy/tinyrenderer/wiki/Lesson-4:-Perspective-projection#homogeneous-coordinates

/**
  * Matrix in the shape of (use of homogeneous coordinates) :
  * c00 c01 c02 c03
  * c10 c11 c12 c13
  * c20 c21 c22 c23
  *   0   0   0   1
  *
  * @param content the content of the matrix
  */
class Matrix(val content : Seq[Seq[Double]]) {

  /**
    * Computes the product between a point P(x ; y ; z) and the matrix.
    *
    * @param point a point P(x ; y ; z ; 1)
    * @return a new point P'(
    *         x * c00 + y * c10 + z * c20
    *         ;
    *         x * c01 + y * c11 + z * c21
    *         ;
    *         x * c02 + y * c12 + z * c22
    *         ;
    *         1
    *         )
    */
  def product(point : Seq[Double]) : Seq[Double] = {
    (0 to 3).map(
      i => content(i).zip(point).map(couple2 => couple2._1 * couple2._2).sum
    )
  }

}

Then, use of similar triangles

Ref. 1/2 : Part. "Of the Importance of Converting Points to Camera Space " of https://www.scratchapixel.com/lessons/3d-basic-rendering/computing-pixel-coordinates-of-3d-point/mathematics-computing-2d-coordinates-of-3d-points

Ref. 2/2 : https://github.com/ssloy/tinyrenderer/wiki/Lesson-4:-Perspective-projection#time-to-work-in-full-3d

NB : at this step, the entries are points expressed according to the camera (i.e. : they are the result of the precedently defined product with the precedently defined matrix).

class Projector {

  /**
    * Computes the coordinates of the projection of the point P on the canvas.
    * The canvas is assumed to be 1 unit forward the camera.
    * The computation uses the definition of the similar triangles.
    *
    * @param points the point P we want to project on the canvas. Its coordinates must be expressed in the coordinates
    *          system of the camera before using this function.
    * @return the point P', projection of P.
    */
  def drawPointsOnCanvas(points : Seq[Seq[Double]]) : Seq[Seq[Double]] = {
    points.map(point => {
      point.map(coordinate => {
        coordinate / point(3)
      }).dropRight(1)
    })

  }

}

Finally, the drawing of the projected points, onto the canvas.

Ref. : Part. "From Screen Space to Raster Space" of https://www.scratchapixel.com/lessons/3d-basic-rendering/computing-pixel-coordinates-of-3d-point/mathematics-computing-2d-coordinates-of-3d-points

import java.awt.Graphics
import javax.swing.JFrame

/**
  * Assumed to be 1 unit forward the camera.
  * Contains the drawn points.
  */
class Canvas(val drawn_points : Seq[Seq[Double]]) extends JFrame {

  val CANVAS_WIDTH = 820
  val CANVAS_HEIGHT = 820
  val IMAGE_WIDTH = 900
  val IMAGE_HEIGHT = 900

  def display = {
    setTitle("Perlin")
    setSize(IMAGE_WIDTH, IMAGE_HEIGHT)
    setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE)
    setVisible(true)
  }

  override def paint(graphics : Graphics): Unit = {
    super.paint(graphics)
    drawn_points.foreach(point => {

      if(!(Math.abs(point.head) <= CANVAS_WIDTH / 2 || Math.abs(point(1)) <= CANVAS_HEIGHT / 2)) {
        println("WARNING : the point (" + point.head + " ; " + point(1) + ") can't be drawn in this canvas.")
      } else {
        val normalized_drawn_point = Seq((point.head + (CANVAS_WIDTH / 2)) / CANVAS_WIDTH, (point(1) + (CANVAS_HEIGHT / 2)) / CANVAS_HEIGHT)
        graphics.fillRect((normalized_drawn_point.head * IMAGE_WIDTH).toInt, ((1 - normalized_drawn_point(1)) * IMAGE_HEIGHT).toInt, 5, 5)

        graphics.drawString(
          "P(" + (normalized_drawn_point.head * IMAGE_WIDTH).toInt + " ; "
          + ((1 - normalized_drawn_point(1)) * IMAGE_HEIGHT).toInt + ")",
          (normalized_drawn_point.head * IMAGE_WIDTH).toInt - 50, ((1 - normalized_drawn_point(1)) * IMAGE_HEIGHT).toInt - 10
        )
      }
    })
  }

}

Question

What's wrong with my program ? I understood the geometrical concepts explained by these both tutorials that I read carefully. I'm pretty sure my product works. I think either the rasterization, or the entries (the matrix) could be wrong...

回答1:

Note that I'm using the simplest perspective projection matrix : I don't use concept of fov, near and far clipping planes.

I think that your projection matrix is too simple. By dropping the near and far clipping planes, you are dropping perspective projection entirely.

You do not have to perform the z-clipping step, but you need to define a view frustum to get perspective to work. I believe that your projection matrix defines a cubic "view frustrum", hence no perspective.

See http://www.songho.ca/opengl/gl_projectionmatrix.html for a discussion of how the projection matrix works.

回答2:

Quoting the Scratchapixel page:

... If we substitute these numbers in the above equation, we get:

Where y' is the y coordinate of P'. Thus:

This is probably one the simplest and most fundamental relation in computer graphics, known as the z or perspective divide. The exact same principle applies to the x coordinate. ...

And in your code:

def drawPointsOnCanvas(points : Seq[Seq[Double]]) : Seq[Seq[Double]] = {
    points.map(point => {
      point.map(coordinate => {
        coordinate / point(3)
                     ^^^^^^^^
    ...

The (3) index is the 4th component of point, i.e. its W-coordinate, not its Z-coordinate. Perhaps you meant coordinate / point(2)?