PyOpenGL 将视图坐标转换为用于 ArcBall 导航的对象坐标
PyOpenGL transforming view coordinates into object coordinates for ArcBall navigation
我正在学习 3d 轨迹球导航教程：
https://en.wikibooks.org/wiki/OpenGL_Programming/Modern_OpenGL_Tutorial_Arcball
我设法执行了所有步骤并且导航正常，但我似乎无法理解教程中的最后一步：
An extra trick is converting the rotation axis from camera coordinates
to object coordinates. It's useful when the camera and object are
placed differently. For instace, if you rotate the object by 90° on
the Y axis ("turn its head" to the right), then perform a vertical
move with your mouse, you make a rotation on the camera X axis, but it
should become a rotation on the Z axis (plane barrel roll) for the
object. By converting the axis in object coordinates, the rotation
will respect that the user work in camera coordinates (WYSIWYG). To
transform from camera to object coordinates, we take the inverse of
the MV matrix (from the MVP matrix triplet).
问题是，当我在第一步中转动模型时，旋转轴也进行了变换，但它们没有与我的 "camera view" 对齐。当然，我想让我的旋转轴始终与我的相机视图对齐。
有人可以给我建议如何解决吗？在教程中有一段代码，但没有太多解释它实际在做什么，而且我只说 Python.
谢谢，
雅各布
我的代码：
import pygame
from pygame.locals import *
from OpenGL.GL import *
from OpenGL.GLU import *
import math
import os
import numpy as np
size = 30
speed = 500
amplitude_amplificator = 80
color_table = ((1,0,0),
(0,1,0),
(0,0,1),
(1,1,0),
(1,0,1),
(0,1,1),
(1,0.5,0),
(0.5,1,0),
(0.5,1,0.5),
(0,0.5,0)
)
locations = ((0,-975, 0),
(0, 975, 0),
(-1273,-975, 0),
(-1273, 975, 0),
(-2482, -975, 0),
(-2482, 975, 0),
(-3737, -975, 0),
(-3737, 975, 0)
)
lines = ((0,2),
(2, 4),
(4, 6),
(1, 3),
(3, 5),
(5, 7),
(0, 1),
(2, 3),
(4, 5),
(6, 7),
)
amplitudes = ((3.38829249165602, 2.38305866657961, 2.52151563664636),
(5.08487438107113, 2.36432294667884, 3.0843991148654),
(3.44312569856563, 1.23112415468012, 1.29869765112226),
(4.0421066637935, 1.40655294535107, 1.36083778879317),
(3.78074337117764, 0.648255908566916, 0.752239154016233),
(5.08887133464996, 0.607037324785205, 0.543523234321567),
(4.49095206021647, 0.432732677308301, 2.18289872563964),
(5.14707697114171, 0.335119576625248, 2.15666871777855)
)
phases = ((-146.873017352057,0,-95.316526141321),
(-149.008372080797, 5.24886681104675, 78.3075732082314),
(-148.241584335287, 5.54327579087787, -118.279685417256),
(-151.844141596427, 6.48705235395368, -113.246406750217),
(-148.14233553496, 27.9523171503408, 65.8254568277543),
(-157.058723259828, 38.8760924034639, 85.2339573112435),
(-153.417593784393, -120.329988461629, 16.0421535833842),
(-156.779107376825, 83.2350395893582, 10.7592173681729)
)
# DRAW CUBE
def Cube(po,si,co):
POS = (
(po[0]+si, po[1]-si, po[2]-si),
(po[0]+si, po[1]+si, po[2]-si),
(po[0]-si, po[1]+si, po[2]-si),
(po[0]-si, po[1]-si, po[2]-si),
(po[0]+si, po[1]-si, po[2]+si),
(po[0]+si, po[1]+si, po[2]+si),
(po[0]-si, po[1]-si, po[2]+si),
(po[0]-si, po[1]+si, po[2]+si)
)
edges = (
(0,1),
(0,3),
(0,4),
(2,1),
(2,3),
(2,7),
(6,3),
(6,4),
(6,7),
(5,1),
(5,4),
(5,7)
)
glBegin(GL_LINES)
for edge in edges:
for vertex in edge:
glColor3f(co[0],co[1],co[2])
glVertex3fv(POS[vertex])
glEnd()
#DRAW ORIGINAL SHAPE IN LINES
def Line_orig(po):
glBegin(GL_LINES)
for edge in po:
for vertex in edge:
glVertex3fv(locations[vertex])
glEnd()
#Hemisphere mapping
def map_hemisphere(x,y):
z = math.sqrt(abs(1-math.pow(x,2)-math.pow(y,2)))
return z
# Calculate angle of two spatial vectors
def angle_calculation(a,b):
r = math.degrees(math.acos((np.dot(a, b))/(np.linalg.norm(a)*np.linalg.norm(b))))
return r
def main():
mouse_pressed = 0
pygame.init()
display = (1200,800)
pygame.display.set_mode(display, DOUBLEBUF|OPENGL)
gluPerspective(45, (display[0]/display[1]), 0.1, 30000.0)
glTranslatef(0,0.0,-10000)
#glRotatef(90, 1, 0, 0)
while True:
for event in pygame.event.get():
if event.type == pygame.QUIT:
pygame.quit()
quit()
glClear(GL_COLOR_BUFFER_BIT|GL_DEPTH_BUFFER_BIT)
time = pygame.time.get_ticks()/1000
norm_mouse_pos = (2*pygame.mouse.get_pos()[0]/display[0]-1,2*pygame.mouse.get_pos()[1]/display[1]-1,map_hemisphere(2*pygame.mouse.get_pos()[0]/display[0]-1,2*pygame.mouse.get_pos()[1]/display[1]-1))
if pygame.mouse.get_pressed()[0]==1:
if mouse_pressed == 0:
mouse_pressed = 1
clear = lambda: os.system('cls')
clear()
p1 = (norm_mouse_pos[0],norm_mouse_pos[1],map_hemisphere(norm_mouse_pos[0],norm_mouse_pos[1]))
print(p1)
else:
p2 = (norm_mouse_pos[0],norm_mouse_pos[1],map_hemisphere(norm_mouse_pos[0],norm_mouse_pos[1]))
cist = np.cross(p1, p2)
print(angle_calculation(p1,p2))
glRotatef( angle_calculation(p1,p2) , -cist[0] , cist[1] , cist[2] )
else:
mouse_pressed = 0
# Translation of the model via keyboard handling
keys=pygame.key.get_pressed()
if keys[K_w]:
glTranslatef(0, 100, 0)
if keys[K_s]:
glTranslatef(0, -100, 0)
if keys[K_a]:
glTranslatef(-100, 0, 0)
if keys[K_d]:
glTranslatef(100, 0, 0)
# Drawing the Cubes at Nodes Loactions
for item, el in enumerate(locations):
Cube((el[0] + amplitudes[item][0]*math.sin(time + phases[item][0]*(3.1415927/180))*amplitude_amplificator,
el[1] + amplitudes[item][1]*math.sin(time + phases[item][1]*(3.1415927/180))*amplitude_amplificator,
el[2] + amplitudes[item][2]*math.sin(time + phases[item][2]*(3.1415927/180))*amplitude_amplificator
), size, color_table[item])
# Drawing the Original Shapes (Specified nodes in Lines Tuple)
Line_orig(lines)
# Drawing the Deformed Shape
glBegin(GL_LINES)
for edge in lines:
for vertex in edge:
glVertex3fv((locations[vertex][0] + amplitudes[vertex][0]*math.sin(time + phases[vertex][0]*(3.1415927/180))*amplitude_amplificator,
locations[vertex][1] + amplitudes[vertex][1]*math.sin(time + phases[vertex][1]*(3.1415927/180))*amplitude_amplificator ,
locations[vertex][2] + amplitudes[vertex][2]*math.sin(time + phases[vertex][2]*(3.1415927/180))*amplitude_amplificator,
))
glEnd()
# OpenGL Management
pygame.display.flip()
pygame.time.wait(10)
main()
The problem is that when i turn the model in the first step axis of rotation transform as well and they are not aligned with my "camera view". Of course I would like to keep my rotation axes always aligned with my camera view.
在渲染中，场景的每个网格通常由模型矩阵、视图矩阵和投影矩阵进行变换。
投影矩阵：
投影矩阵描述了从场景的 3D 点到视口的 2D 点的映射。
查看矩阵：
视图矩阵描述了观察场景的方向和位置。视图矩阵从世界 space 转换为视图（眼睛）space。
模型矩阵：
模型矩阵定义场景中网格的位置、方向和相对大小。模型矩阵将顶点位置从网格变换到世界 space.
如果您想围绕视图 space 中的轴旋转 szene，则必须执行以下操作：
通过在新的旋转操作之前完成的所有旋转和平移来转换模型。
应用新的旋转操作。
应用视图翻译
应用投影矩阵
大小 OpenGL 固定功能管道有一个矩阵堆栈，这个操作必须以相反的顺序进行。
例如请参阅 glMultMatrix:
的文档
glMultMatrix multiplies the current matrix with the one specified using m, and replaces the current matrix with the product.
在 OpenGL 中，每种矩阵模式都有一个矩阵堆栈（参见 glMatrixMode）。矩阵模式为 GL_MODELVIEW、GL_PROJECTION 和 GL_TEXTURE.
首先你必须在分离的投影矩阵堆栈上设置投影矩阵：
glMatrixMode( GL_PROJECTION );
gluPerspective(45, (display[0]/display[1]), 0.1, 30000.0)
接下来创建模型矩阵
a = (GLfloat * 16)()
modelMat = glGetFloatv(GL_MODELVIEW_MATRIX, a)
在主循环中初始化模型视图矩阵：
glMatrixMode( GL_MODELVIEW );
glLoadIdentity()
计算新的旋转和平移：
axis = (p2[0]- p1[0], p2[1]- p1[1])
glRotatef( angle_calculation(p1,p2), axis[1], axis[0], 0 )
将模型矩阵乘以先前的模型矩阵并存储组合模型矩阵：
glMultMatrixf( modelMat )
modelMat = glGetFloatv(GL_MODELVIEW_MATRIX, a)
设置视图并应用新模型矩阵：
glLoadIdentity()
glTranslatef(0,0.0,-10000)
glMultMatrixf( modelMat )
最终代码可能如下所示：
.....
glMatrixMode( GL_PROJECTION );
gluPerspective(45, (display[0]/display[1]), 0.1, 30000.0)
a = (GLfloat * 16)()
modelMat = glGetFloatv(GL_MODELVIEW_MATRIX, a)
while True:
for event in pygame.event.get():
if event.type == pygame.QUIT:
pygame.quit()
quit()
glMatrixMode( GL_MODELVIEW );
glLoadIdentity()
norm_mouse_pos = (2*pygame.mouse.get_pos()[0]/display[0]-1,2*pygame.mouse.get_pos()[1]/display[1]-1,map_hemisphere(2*pygame.mouse.get_pos()[0]/display[0]-1,2*pygame.mouse.get_pos()[1]/display[1]-1))
if pygame.mouse.get_pressed()[0]==1:
if mouse_pressed == 0:
mouse_pressed = 1
clear = lambda: os.system('cls')
clear()
p1 = (norm_mouse_pos[0],norm_mouse_pos[1],map_hemisphere(norm_mouse_pos[0],norm_mouse_pos[1]))
else:
p2 = (norm_mouse_pos[0],norm_mouse_pos[1],map_hemisphere(norm_mouse_pos[0],norm_mouse_pos[1]))
cist = np.cross(p1, p2)
axis = (p2[0]- p1[0], p2[1]- p1[1])
glRotatef( angle_calculation(p1,p2) , axis[1] , axis[0] , 0 )
else:
mouse_pressed = 0
# Translation of the model via keyboard handling
keys=pygame.key.get_pressed()
if keys[K_w]:
glTranslatef(0, 100, 0)
if keys[K_s]:
glTranslatef(0, -100, 0)
if keys[K_a]:
glTranslatef(-100, 0, 0)
if keys[K_d]:
glTranslatef(100, 0, 0)
glMultMatrixf( modelMat )
modelMat = glGetFloatv(GL_MODELVIEW_MATRIX, a)
glLoadIdentity()
glTranslatef(0,0.0,-10000)
glMultMatrixf( modelMat )
glClear(GL_COLOR_BUFFER_BIT|GL_DEPTH_BUFFER_BIT)
.....
有时会出现 "ValueError: math domain error"，这是因为仅当值在 [-1, 1] 范围内时才定义值的反余弦。将该值限制在该范围内 (min(1,max(cos_a,-1)))：
def angle_calculation(a,b):
cos_a = np.dot(a, b) / (np.linalg.norm(a)*np.linalg.norm(b))
r = math.degrees(math.acos( min(1,max(cos_a,-1)) ))
return r
我正在学习 3d 轨迹球导航教程：
https://en.wikibooks.org/wiki/OpenGL_Programming/Modern_OpenGL_Tutorial_Arcball
我设法执行了所有步骤并且导航正常，但我似乎无法理解教程中的最后一步：
An extra trick is converting the rotation axis from camera coordinates to object coordinates. It's useful when the camera and object are placed differently. For instace, if you rotate the object by 90° on the Y axis ("turn its head" to the right), then perform a vertical move with your mouse, you make a rotation on the camera X axis, but it should become a rotation on the Z axis (plane barrel roll) for the object. By converting the axis in object coordinates, the rotation will respect that the user work in camera coordinates (WYSIWYG). To transform from camera to object coordinates, we take the inverse of the MV matrix (from the MVP matrix triplet).
问题是，当我在第一步中转动模型时，旋转轴也进行了变换，但它们没有与我的 "camera view" 对齐。当然，我想让我的旋转轴始终与我的相机视图对齐。
有人可以给我建议如何解决吗？在教程中有一段代码，但没有太多解释它实际在做什么，而且我只说 Python.
谢谢， 雅各布
我的代码：
import pygame
from pygame.locals import *
from OpenGL.GL import *
from OpenGL.GLU import *
import math
import os
import numpy as np
size = 30
speed = 500
amplitude_amplificator = 80
color_table = ((1,0,0),
(0,1,0),
(0,0,1),
(1,1,0),
(1,0,1),
(0,1,1),
(1,0.5,0),
(0.5,1,0),
(0.5,1,0.5),
(0,0.5,0)
)
locations = ((0,-975, 0),
(0, 975, 0),
(-1273,-975, 0),
(-1273, 975, 0),
(-2482, -975, 0),
(-2482, 975, 0),
(-3737, -975, 0),
(-3737, 975, 0)
)
lines = ((0,2),
(2, 4),
(4, 6),
(1, 3),
(3, 5),
(5, 7),
(0, 1),
(2, 3),
(4, 5),
(6, 7),
)
amplitudes = ((3.38829249165602, 2.38305866657961, 2.52151563664636),
(5.08487438107113, 2.36432294667884, 3.0843991148654),
(3.44312569856563, 1.23112415468012, 1.29869765112226),
(4.0421066637935, 1.40655294535107, 1.36083778879317),
(3.78074337117764, 0.648255908566916, 0.752239154016233),
(5.08887133464996, 0.607037324785205, 0.543523234321567),
(4.49095206021647, 0.432732677308301, 2.18289872563964),
(5.14707697114171, 0.335119576625248, 2.15666871777855)
)
phases = ((-146.873017352057,0,-95.316526141321),
(-149.008372080797, 5.24886681104675, 78.3075732082314),
(-148.241584335287, 5.54327579087787, -118.279685417256),
(-151.844141596427, 6.48705235395368, -113.246406750217),
(-148.14233553496, 27.9523171503408, 65.8254568277543),
(-157.058723259828, 38.8760924034639, 85.2339573112435),
(-153.417593784393, -120.329988461629, 16.0421535833842),
(-156.779107376825, 83.2350395893582, 10.7592173681729)
)
# DRAW CUBE
def Cube(po,si,co):
POS = (
(po[0]+si, po[1]-si, po[2]-si),
(po[0]+si, po[1]+si, po[2]-si),
(po[0]-si, po[1]+si, po[2]-si),
(po[0]-si, po[1]-si, po[2]-si),
(po[0]+si, po[1]-si, po[2]+si),
(po[0]+si, po[1]+si, po[2]+si),
(po[0]-si, po[1]-si, po[2]+si),
(po[0]-si, po[1]+si, po[2]+si)
)
edges = (
(0,1),
(0,3),
(0,4),
(2,1),
(2,3),
(2,7),
(6,3),
(6,4),
(6,7),
(5,1),
(5,4),
(5,7)
)
glBegin(GL_LINES)
for edge in edges:
for vertex in edge:
glColor3f(co[0],co[1],co[2])
glVertex3fv(POS[vertex])
glEnd()
#DRAW ORIGINAL SHAPE IN LINES
def Line_orig(po):
glBegin(GL_LINES)
for edge in po:
for vertex in edge:
glVertex3fv(locations[vertex])
glEnd()
#Hemisphere mapping
def map_hemisphere(x,y):
z = math.sqrt(abs(1-math.pow(x,2)-math.pow(y,2)))
return z
# Calculate angle of two spatial vectors
def angle_calculation(a,b):
r = math.degrees(math.acos((np.dot(a, b))/(np.linalg.norm(a)*np.linalg.norm(b))))
return r
def main():
mouse_pressed = 0
pygame.init()
display = (1200,800)
pygame.display.set_mode(display, DOUBLEBUF|OPENGL)
gluPerspective(45, (display[0]/display[1]), 0.1, 30000.0)
glTranslatef(0,0.0,-10000)
#glRotatef(90, 1, 0, 0)
while True:
for event in pygame.event.get():
if event.type == pygame.QUIT:
pygame.quit()
quit()
glClear(GL_COLOR_BUFFER_BIT|GL_DEPTH_BUFFER_BIT)
time = pygame.time.get_ticks()/1000
norm_mouse_pos = (2*pygame.mouse.get_pos()[0]/display[0]-1,2*pygame.mouse.get_pos()[1]/display[1]-1,map_hemisphere(2*pygame.mouse.get_pos()[0]/display[0]-1,2*pygame.mouse.get_pos()[1]/display[1]-1))
if pygame.mouse.get_pressed()[0]==1:
if mouse_pressed == 0:
mouse_pressed = 1
clear = lambda: os.system('cls')
clear()
p1 = (norm_mouse_pos[0],norm_mouse_pos[1],map_hemisphere(norm_mouse_pos[0],norm_mouse_pos[1]))
print(p1)
else:
p2 = (norm_mouse_pos[0],norm_mouse_pos[1],map_hemisphere(norm_mouse_pos[0],norm_mouse_pos[1]))
cist = np.cross(p1, p2)
print(angle_calculation(p1,p2))
glRotatef( angle_calculation(p1,p2) , -cist[0] , cist[1] , cist[2] )
else:
mouse_pressed = 0
# Translation of the model via keyboard handling
keys=pygame.key.get_pressed()
if keys[K_w]:
glTranslatef(0, 100, 0)
if keys[K_s]:
glTranslatef(0, -100, 0)
if keys[K_a]:
glTranslatef(-100, 0, 0)
if keys[K_d]:
glTranslatef(100, 0, 0)
# Drawing the Cubes at Nodes Loactions
for item, el in enumerate(locations):
Cube((el[0] + amplitudes[item][0]*math.sin(time + phases[item][0]*(3.1415927/180))*amplitude_amplificator,
el[1] + amplitudes[item][1]*math.sin(time + phases[item][1]*(3.1415927/180))*amplitude_amplificator,
el[2] + amplitudes[item][2]*math.sin(time + phases[item][2]*(3.1415927/180))*amplitude_amplificator
), size, color_table[item])
# Drawing the Original Shapes (Specified nodes in Lines Tuple)
Line_orig(lines)
# Drawing the Deformed Shape
glBegin(GL_LINES)
for edge in lines:
for vertex in edge:
glVertex3fv((locations[vertex][0] + amplitudes[vertex][0]*math.sin(time + phases[vertex][0]*(3.1415927/180))*amplitude_amplificator,
locations[vertex][1] + amplitudes[vertex][1]*math.sin(time + phases[vertex][1]*(3.1415927/180))*amplitude_amplificator ,
locations[vertex][2] + amplitudes[vertex][2]*math.sin(time + phases[vertex][2]*(3.1415927/180))*amplitude_amplificator,
))
glEnd()
# OpenGL Management
pygame.display.flip()
pygame.time.wait(10)
main()
The problem is that when i turn the model in the first step axis of rotation transform as well and they are not aligned with my "camera view". Of course I would like to keep my rotation axes always aligned with my camera view.
在渲染中，场景的每个网格通常由模型矩阵、视图矩阵和投影矩阵进行变换。
投影矩阵：
投影矩阵描述了从场景的 3D 点到视口的 2D 点的映射。查看矩阵：
视图矩阵描述了观察场景的方向和位置。视图矩阵从世界 space 转换为视图（眼睛）space。模型矩阵：
模型矩阵定义场景中网格的位置、方向和相对大小。模型矩阵将顶点位置从网格变换到世界 space.
如果您想围绕视图 space 中的轴旋转 szene，则必须执行以下操作：
通过在新的旋转操作之前完成的所有旋转和平移来转换模型。
应用新的旋转操作。
应用视图翻译
应用投影矩阵
大小 OpenGL 固定功能管道有一个矩阵堆栈，这个操作必须以相反的顺序进行。
例如请参阅 glMultMatrix:
glMultMatrixmultiplies the current matrix with the one specified usingm, and replaces the current matrix with the product.
在 OpenGL 中，每种矩阵模式都有一个矩阵堆栈（参见 glMatrixMode）。矩阵模式为 GL_MODELVIEW、GL_PROJECTION 和 GL_TEXTURE.
首先你必须在分离的投影矩阵堆栈上设置投影矩阵：
glMatrixMode( GL_PROJECTION );
gluPerspective(45, (display[0]/display[1]), 0.1, 30000.0)
接下来创建模型矩阵
a = (GLfloat * 16)()
modelMat = glGetFloatv(GL_MODELVIEW_MATRIX, a)
在主循环中初始化模型视图矩阵：
glMatrixMode( GL_MODELVIEW );
glLoadIdentity()
计算新的旋转和平移：
axis = (p2[0]- p1[0], p2[1]- p1[1])
glRotatef( angle_calculation(p1,p2), axis[1], axis[0], 0 )
将模型矩阵乘以先前的模型矩阵并存储组合模型矩阵：
glMultMatrixf( modelMat )
modelMat = glGetFloatv(GL_MODELVIEW_MATRIX, a)
设置视图并应用新模型矩阵：
glLoadIdentity()
glTranslatef(0,0.0,-10000)
glMultMatrixf( modelMat )
最终代码可能如下所示：
.....
glMatrixMode( GL_PROJECTION );
gluPerspective(45, (display[0]/display[1]), 0.1, 30000.0)
a = (GLfloat * 16)()
modelMat = glGetFloatv(GL_MODELVIEW_MATRIX, a)
while True:
for event in pygame.event.get():
if event.type == pygame.QUIT:
pygame.quit()
quit()
glMatrixMode( GL_MODELVIEW );
glLoadIdentity()
norm_mouse_pos = (2*pygame.mouse.get_pos()[0]/display[0]-1,2*pygame.mouse.get_pos()[1]/display[1]-1,map_hemisphere(2*pygame.mouse.get_pos()[0]/display[0]-1,2*pygame.mouse.get_pos()[1]/display[1]-1))
if pygame.mouse.get_pressed()[0]==1:
if mouse_pressed == 0:
mouse_pressed = 1
clear = lambda: os.system('cls')
clear()
p1 = (norm_mouse_pos[0],norm_mouse_pos[1],map_hemisphere(norm_mouse_pos[0],norm_mouse_pos[1]))
else:
p2 = (norm_mouse_pos[0],norm_mouse_pos[1],map_hemisphere(norm_mouse_pos[0],norm_mouse_pos[1]))
cist = np.cross(p1, p2)
axis = (p2[0]- p1[0], p2[1]- p1[1])
glRotatef( angle_calculation(p1,p2) , axis[1] , axis[0] , 0 )
else:
mouse_pressed = 0
# Translation of the model via keyboard handling
keys=pygame.key.get_pressed()
if keys[K_w]:
glTranslatef(0, 100, 0)
if keys[K_s]:
glTranslatef(0, -100, 0)
if keys[K_a]:
glTranslatef(-100, 0, 0)
if keys[K_d]:
glTranslatef(100, 0, 0)
glMultMatrixf( modelMat )
modelMat = glGetFloatv(GL_MODELVIEW_MATRIX, a)
glLoadIdentity()
glTranslatef(0,0.0,-10000)
glMultMatrixf( modelMat )
glClear(GL_COLOR_BUFFER_BIT|GL_DEPTH_BUFFER_BIT)
.....
有时会出现 "ValueError: math domain error"，这是因为仅当值在 [-1, 1] 范围内时才定义值的反余弦。将该值限制在该范围内 (min(1,max(cos_a,-1)))：
def angle_calculation(a,b):
cos_a = np.dot(a, b) / (np.linalg.norm(a)*np.linalg.norm(b))
r = math.degrees(math.acos( min(1,max(cos_a,-1)) ))
return r