Affine transformation matrix offset
This has been killing me the last few days. Not even kidding, but I've been really stressing over this trying to solve it.
I am currently trying to use affine transformation matrices to create an isometric projection in HTML5. I receive a tile which is a square that is rotated 45 degrees (essentially a square diamond on a square canvas). I then scale one of the axis' depending on if the there is a delta in the x or y direction. I then skew the axis by a factor to fit. Then, I negate the initial rotation by rotating it back by -45 degrees.
Currently, my affine matrix is:
// note: the difference in z is about 10 in this example,
// so, xDiff is usually 40
var xDiff = 4 * (center.z - map[x+1][y].land.z);
var yDiff = 4 * (center.z - map[x][y+1].land.z);
var matrix = multiplyAll(
// Rotation
[COS45, SIN45,
-SIN45, COS45],
// Scale in each respective axis
[(44+yDiff)/44, 0,
0, (44+xDiff)/44],
// Skew each axis
[1, -yDiff/(44+yDiff),
-xDiff/(44+xDiff), 1],
// Negate the rotation
[NCOS45, NSIN45,
-NSIN45, NCOS45]
);
Then I draw it using:
// the map has its own x & y values which directions are determined by the red x & y arrows in the picture
// pX & pY are the point relative to the canvas origin
var pX = x * 22 - y * 22 + 22;
var pY = y * 22 + x * 22 - 22 - (center.z * 4);
context.setTransform(matrix[0], matrix[1],
matrix[2], matrix[3],
300, 100);
//m_Context.drawImage(image, pX, pY);
drawDiamond(pX, pY, true); // draws a 44x44 diamond
As you can see, the transformed matrices are being drawn with respect to the transformed x-axis (I think the "new" x-axis has a slope of yDiff/44). I'm not sure how to draw the shapes so that the transformed result will be in the correct position. Using pY = x * 22 - (yDiff/10);
seems to get the point closer, but I pretty much guessed it by plugging in random numbers.
tl;dr:
PS: The weird diamonds on the bottom can be ignored for now since they can correctly be created ONCE I find out how to calculate the offsets.
An affine transformation matrix ([abcdef]) expresses the two equations
x' = ax + cy + e
y' = bx + dy + f
So, you can use the offsets e and f to bypass the scaling and skewing parts (the 4x4 linear transform embedded in the 2x3 or 3x3 matrix).
This is used a lot in postscript programming, where the coordinates used for drawing an object are relative to a local origin. If you're concatenating matrices, do the translation before scaling and skewing and the e and f values will remain unmolested.
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