c++ - Opengl:保持 Arcball 相机向上 vector 与 y 轴对齐

Question

我基本上是在尝试模仿相机在 Maya 中的旋转方式。 Maya 中的轨迹球始终与 y 轴对齐。因此，无论向上 vector 指向何处，它仍然会沿 y 轴旋转或与向上 vector 对齐。

我已经能够使用 C++ 和 Qt 在 OpenGL 中实现 arcball。但我不知道如何保持它的向上 vector 对齐。我已经能够通过下面的代码使其有时保持对齐:

void ArcCamera::setPos (Vector3 np)
{
    Vector3 up(0, 1, 0);

    Position = np;
    ViewDir = (ViewPoint - Position); ViewDir.normalize();

    RightVector =  ViewDir ^ up; RightVector.normalize();

    UpVector = RightVector ^ ViewDir;  UpVector.normalize();
}

这一直有效，直到位置达到 90 度，然后右 vector 发生变化，一切都反转了。

因此，我一直在保持总旋转(四元数)并按它旋转原始位置(上、右、位置)。这最适合保持一切连贯，但现在我根本无法将向上 vector 与 y 轴对齐。下面是旋转的函数。

void CCamera::setRot (QQuaternion q)
{
    tot = tot * q;

    Position  = tot.rotatedVector(PositionOriginal);

    UpVector = tot.rotatedVector(UpVectorOriginal);
    UpVector.normalize();

    RightVector  = tot.rotatedVector(RightVectorOriginal);
    RightVector.normalize();
}

QQuaternion q 是从鼠标拖动导出的轴角对生成的。我相信这是正确完成的。旋转本身很好，只是不能保持方向对齐。

我注意到在我选择的实现中，在角中拖动提供了围绕我的 View 方向的旋转，而且我始终可以重新对齐向上 vector 以拉直到世界的 y 轴方向。所以如果我能弄清楚要滚动多少，我可能每次都旋转两次以确保它是直的。但是，我不确定该怎么做。

Answer 1

这不起作用的原因是 Maya 在视口(viewport)中的相机操作不使用 arcball 界面。你要做的是Maya's tumble command .我找到的最好的解释资源是 this document from Professor Orr's Computer Graphics class .

鼠标左右移动对应方位角，指定绕世界空间Y轴旋转。上下移动鼠标对应于仰角，并指定围绕 View 空间 X 轴的旋转。目标是生成新的 world-to-view 矩阵，然后根据您对相机的参数化方式从该矩阵中提取新的相机方向和眼睛位置。

从当前的世界到 View 矩阵开始。接下来，我们需要在世界空间中定义枢轴点。任何枢轴点都可以开始，使用世界原点可能是最简单的。

回想一下，纯旋转矩阵生成以原点为中心的旋转。这意味着要围绕任意轴心点旋转，您首先要平移到原点，执行旋转，然后平移回来。还请记住，变换组合是从右到左发生的，因此到达原点的负平移发生在最右边:

translate(pivotPosition) * rotate(angleX, angleY, angleZ) * translate(-pivotPosition)

我们可以用它来计算方位角旋转分量，它是围绕世界 Y 轴的旋转:

azimuthRotation = translate(pivotPosition) * rotateY(angleY) * translate(-pivotPosition)

我们必须为仰角旋转组件做一些额外的工作，因为它发生在 View 空间中，围绕 View 空间 X 轴:

elevationRotation = translate(worldToViewMatrix * pivotPosition) * rotateX(angleX) * translate(worldToViewMatrix * -pivotPosition)

然后我们可以获得新的 View 矩阵:

newWorldToViewMatrix = elevationRotation * worldToViewMatrix * azimuthRotation

现在我们有了新的 worldToView 矩阵，剩下的就是必须从 View 矩阵中提取新的世界空间位置和方向。为此，我们需要 viewToWorld 矩阵，它是 worldToView 矩阵的逆矩阵。

newOrientation = transpose(mat3(newWorldToViewMatrix))
newPosition = -((newOrientation * newWorldToViewMatrix).column(3))

此时，我们已经将元素分开了。如果你的相机被参数化，那么你只需要为你的方向存储一个四元数，你只需要做旋转矩阵 ->四元数转换。当然，Maya 会转换为欧拉角显示在 channel 盒中，这将取决于相机的旋转顺序(请注意，当旋转顺序改变时，翻滚的数学不会改变，只是旋转的方式矩阵 -> 欧拉角转换完成)​​。

这是 Python 中的示例实现:

#!/usr/bin/env python

import numpy as np
from math import *

def translate(amount):
    'Make a translation matrix, to move by `amount`'
    t = np.matrix(np.eye(4))
    t[3] = amount.T
    t[3, 3] = 1
    return t.T

def rotateX(amount):
    'Make a rotation matrix, that rotates around the X axis by `amount` rads'
    c = cos(amount)
    s = sin(amount)

    return np.matrix([
        [1, 0, 0, 0],
        [0, c,-s, 0],
        [0, s, c, 0],
        [0, 0, 0, 1],
    ])

def rotateY(amount):
    'Make a rotation matrix, that rotates around the Y axis by `amount` rads'
    c = cos(amount)
    s = sin(amount)
    return np.matrix([
        [c, 0, s, 0],
        [0, 1, 0, 0],
       [-s, 0, c, 0],
        [0, 0, 0, 1],
    ])

def rotateZ(amount):
    'Make a rotation matrix, that rotates around the Z axis by `amount` rads'
    c = cos(amount)
    s = sin(amount)
    return np.matrix([
        [c,-s, 0, 0],
        [s, c, 0, 0],
        [0, 0, 1, 0],
        [0, 0, 0, 1],
    ])

def rotate(x, y, z, pivot):
    'Make a XYZ rotation matrix, with `pivot` as the center of the rotation'
    m = rotateX(x) * rotateY(y) * rotateZ(z)

    I = np.matrix(np.eye(4))
    t = (I-m) * pivot
    m[0, 3] = t[0, 0]
    m[1, 3] = t[1, 0]
    m[2, 3] = t[2, 0]

    return m

def eulerAnglesZYX(matrix):
    'Extract the Euler angles from an ZYX rotation matrix'
    x = atan2(-matrix[1, 2], matrix[2, 2])
    cy = sqrt(1 - matrix[0, 2]**2)
    y = atan2(matrix[0, 2], cy)
    sx = sin(x)
    cx = cos(x)
    sz = cx * matrix[1, 0] + sx * matrix[2, 0]
    cz = cx * matrix[1, 1] + sx * matrix[2, 1]
    z = atan2(sz, cz)
    return np.array((x, y, z),)

def eulerAnglesXYZ(matrix):
    'Extract the Euler angles from an XYZ rotation matrix'
    z = atan2(matrix[1, 0], matrix[0, 0])
    cy = sqrt(1 - matrix[2, 0]**2)
    y = atan2(-matrix[2, 0], cy)
    sz = sin(z)
    cz = cos(z)
    sx = sz * matrix[0, 2] - cz * matrix[1, 2]
    cx = cz * matrix[1, 1] - sz * matrix[0, 1]
    x = atan2(sx, cx)
    return np.array((x, y, z),)

class Camera(object):
    def __init__(self, worldPos, rx, ry, rz, coi):
        # Initialize the camera orientation.  In this case the original
        # orientation is built from XYZ Euler angles.  orientation is the top
        # 3x3 XYZ rotation matrix for the view-to-world matrix, and can more
        # easily be thought of as the world space orientation.
        self.orientation = \
            (rotateZ(rz) * rotateY(ry) * rotateX(rx))

        # position is a point in world space for the camera.
        self.position = worldPos

        # Construct the world-to-view matrix, which is the inverse of the
        # view-to-world matrix.
        self.view = self.orientation.T * translate(-self.position)

        # coi is the "center of interest".  It defines a point that is coi
        # units in front of the camera, which is the pivot for the tumble
        # operation.
        self.coi = coi

    def tumble(self, azimuth, elevation):
        '''Tumble the camera around the center of interest.

        Azimuth is the number of radians to rotate around the world-space Y axis.
        Elevation is the number of radians to rotate around the view-space X axis.
        '''
        # Find the world space pivot point.  This is the view position in world
        # space minus the view direction vector scaled by the center of
        # interest distance.
        pivotPos = self.position - (self.coi * self.orientation.T[2]).T

        # Construct the azimuth and elevation transformation matrices
        azimuthMatrix = rotate(0, -azimuth, 0, pivotPos) 
        elevationMatrix = rotate(elevation, 0, 0, self.view * pivotPos)

        # Get the new view matrix
        self.view = elevationMatrix * self.view * azimuthMatrix

        # Extract the orientation from the new view matrix
        self.orientation = np.matrix(self.view).T
        self.orientation.T[3] = [0, 0, 0, 1]

        # Now extract the new view position
        negEye = self.orientation * self.view
        self.position = -(negEye.T[3]).T
        self.position[3, 0] = 1

np.set_printoptions(precision=3)

pos = np.matrix([[5.321, 5.866, 4.383, 1]]).T
orientation = radians(-60), radians(40), 0
coi = 1

camera = Camera(pos, *orientation, coi=coi)
print 'Initial attributes:'
print np.round(np.degrees(eulerAnglesXYZ(camera.orientation)), 3)
print np.round(camera.position, 3)
print 'Attributes after tumbling:'
camera.tumble(azimuth=radians(-40), elevation=radians(-60))
print np.round(np.degrees(eulerAnglesXYZ(camera.orientation)), 3)
print np.round(camera.position, 3)