3d transform

最新推荐文章于 2025-10-17 18:22:41 发布

转载最新推荐文章于 2025-10-17 18:22:41 发布 · 1k 阅读

本文详细介绍了三维空间中平移、缩放及旋转的基本变换矩阵，并分别给出了右手坐标系与左手坐标系下的具体数学表达形式。通过这些变换矩阵，可以实现三维图形的位置调整、大小改变以及方向旋转。

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Translation:
        (1 0 0 0)
    T = (0 1 0 0)
        (0 0 1 0)
        (Tx Ty Tz 1)
    [x' y' z' 1] = [x y z 1].T

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Scaling:
        (Sx 0 0 0)
    S = (0 Sy 0 0)
        (0 0 Sz 0)
        (0 0 0 1)
    [x' y' z' 1] = [x y z 1].S

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Rotation_Right Hand Coordinate:

    by x axis:
                (1   0     0    0)
        Rx(q) = (0 cosq -sinq 0)
                (0 sinq cosq 0)
                (0   0    0     1)
        [x' y' z' 1] = [x y z 1].Rx(q)

        x' = x
        y' = z*sinq + y*cosq
        z' = z*cosq -y*sinq

    by y axis:
                (cosq 0 sinq   0)
        Ry(q) = (0     1   0     0)
                (-sinq 0 cosq   0)
                (0     0   0     1)
        [x' y' z' 1] = [x y z 1].Ry(q)

        x' = x*sinq + z*cosq
        y' = y
        z' = x*cosq - z*sinq

    by z axis:
                (cosq -sinq 0 0)
        Rz(q) = (sinq cosq 0 0)
                (0      0    1 0)
                (0      0    0 1)
        [x' y' z' 1] = [x y z 1].Rz(q)

        x' = y*sinq + x*cosq
        y' = y*cosq - x*sinq
        z' = z

=============================================================
Rotation_Left Hand Coordinate

    by x axis:
                (1   0     0    0)
        Rx(q) = (0 cosq sinq 0)
                (0 -sinq cosq 0)
                (0   0     0    1)
        [x' y' z' 1] = [x y z 1].Rx(q)

        x' = x
        y' = y*cosq - z*sinq
        z' = y*sinq + z*cosq

    by y axis:
                (cosq 0 -sinq 0)
        Ry(q) = (0     1   0    0)
                (sinq 0 cosq   0)
                (0     0   0    1)
        [x' y' z' 1] = [x y z 1].Ry(q)

        x' = z*sinq + x*cosq
        y' = y
        z' = z*cosq - x*sinq

    by z axis:
                (cosq sinq 0 0)
        Rz(q) = (-sinq cosq 0 0)
                (0      0    1 0)
                (0      0    0 1)
        [x' y' z' 1] = [x y z 1].Rz(q)

        x' = x*cosq - y*sinq
        y' = x*sinq + y*cosq
        z' = z