% 北太天元 學(xué)習(xí) opengl 提供的 gluLookAt ,?glMultMatrix,

% glScalef

%

%

%立方體

%

points = [

???% 下底面的四個(gè)點(diǎn)

???-1 -1 -1??; %第一個(gè)點(diǎn)

???-1 1 -1??; %第二個(gè)點(diǎn)

???1 1 -1??;

???1 -1 -1??;

??????%上底面的四個(gè)點(diǎn)

???-1 -1 3??;

???-1 1 3??;

???1 1 3??;

???1 -1 3??;

??????%原點(diǎn)

??????0 0?0

???];

points = points' ; %每個(gè)點(diǎn)的坐標(biāo)改成列向量

%求出所有的點(diǎn)的"中心" center

center = ( max(points,[],2) + min(points, [], 2) ) / 2.0;

%我們就是盯著上面定義的中心點(diǎn)看

target = center;

%把target點(diǎn)平移到原點(diǎn)

points_p = points - target;

%縮放

xyz_max = max(points,[],2); xyz_min = min(points,[], 2);

z_max = xyz_max(3);

z_min = xyz_min(3);

if( z_max == z_min )

???error('所有的點(diǎn)的z坐標(biāo)相等')

end

scale_z = 1.0/(z_max - z_min);

points_s_p = diag([1.0, 1.0, scale_z]) * points_p;?

pitch = 30;

yaw?= 45;

direction = pitch_yaw(pitch, yaw)

up = [0,0,1];

ca_axes = camera_axes(up, direction)

points_r_s_p =?ca_axes' * points_s_p;

ca_min = min(points_r_s_p,[],2)

ca_max = max(points_r_s_p,[],2)

%現(xiàn)在看的方向就是eta軸的正方向

%可以考慮把相機(jī)平移

distance(2) = ca_min(2) - 0.1*(ca_max(2) - ca_min(2))

%相機(jī)在(xi,eta,zeta)下的坐標(biāo)

distance = [0; distance(2); 0 ]; %僅僅考慮在direction方向上的移動(dòng)

%如果假想相機(jī)還在原點(diǎn)，我們移動(dòng)的是觀察的物體，那么觀察物體的移動(dòng)方向和上面

% distance 給出的方向相反

ca_points_1 = points_r_s_p - distance

/**

opengl代碼

glMode(GL_MODELVIEW)

glLoadIdenty()

glTranslatef( 0, -distance(2), 0) ;

glRotaef( yaw ,?.. )

glRotatef( pitch ... )

glScalef(

glTranslatef(?)

*/

%實(shí)際上我們不需要計(jì)算camera的position, 只需要看ca_points_1 是如何計(jì)算出來

% 就知道這么費(fèi)勁算出來的gluLookAt 矩陣實(shí)際上等于

mat2 = [ [ eye(3), [0; -distance(2); 0] ];

?????????[ 0 0 0 1] ] *?...?%平移矩陣

?????????[ [ca_axes', [ 0; 0; 0 ]];

?????????[ 0 0 0 1] ] * ...?%?%旋轉(zhuǎn)矩陣

?????????[ [diag([1.0, 1.0, scale_z]), [ 0; 0; 0 ]];

?????????[ 0 0 0 1] ] * ...?% 縮放矩陣

?????????[ [ eye(3), -target(:) ];

?????????[ 0 0 0 1] ] ;???%平移矩陣

qi_points = [ points ; ones(1, length(points(1,:)))];

ca_points_2 = mat2*qi_points;

%相機(jī)的位置在哪兒呢？ 我們知道通過我們其他渠道求出的LookAt矩陣

% 乘以相機(jī)的位置 = (0,0,0), 于是我們輕松求出相機(jī)的位置

% 下面是齊次坐標(biāo)

camera_position_2 = ( [ [ eye(3), [0; -distance(2); 0] ];

?????????[ 0 0 0 1] ] *?...?%平移矩陣

?????????[ [ca_axes', [ 0; 0; 0 ]];

?????????[ 0 0 0 1] ] ) \ [ 0 ; 0 ; 0; 1 ]

%我們?nèi)绻褂胓luLookAt 就必然要計(jì)算出camera position 和 target

%相機(jī)的方向是我們通過pitch 和 yaw 計(jì)算出來的，

% target 我們假設(shè)是觀察物體的中心，這里就是給出的target

% 需要平移的距離大小是 distance(2), 方向是direction

camera_position = camera_position_2(1:3);

mat = gluLookAt( camera_position, [0,0,0], up )

ca_points_3 =?mat * ( [ [diag([1.0, 1.0, scale_z]), [ 0; 0; 0 ]];

?????????[ 0 0 0 1] ] * ...?% 縮放矩陣

?????????[ [ eye(3), -target(:) ];

?????????[ 0 0 0 1] ] ) * qi_points ;

%???^ z 軸

%???|

%???|?????. y 軸

%???|???.

%???|?.?

%???|_________> x 軸

% 開始的時(shí)候人眼看的方向是 y 軸正向, 然后 pitch 和 yaw

%根據(jù) pitch 和 yaw 確定 camer 的方向

% 開始的camera的方向是(0,1,0), 首先繞x軸正向逆時(shí)針方向旋轉(zhuǎn)pitch角度

% 然后再繞z軸正向逆時(shí)針方向旋轉(zhuǎn) yaw 角度

% 最后得到的camera的方向

% 例如 pitch_yaw(-90,0) 得到的方向是 (0,0,-1);

% 不過，最后的向量都寫成了列向量

function direction = pitch_yaw( pitch, yaw)

???direction = [

?????????-cos( deg2rad(pitch) )*sin( deg2rad(yaw) );

?????????cos( deg2rad(pitch) )*cos( deg2rad(yaw) );

?????????sin( deg2rad(pitch) );

??????];

end

%對(duì)于開始給定的up方向和 direction方向

% 計(jì)算出right 方向 : camera_right = direction x up 再normalize

% 然后根據(jù)camera_right方向和direction方向計(jì)算出camera_up:

% camera_up = right x direction?再normalize

% 最后得到的camera坐標(biāo)系的三個(gè)軸是

%?(right, direction, up)

% 注意: https://learnopengl.com/Getting-started/Camera 這篇文章中的

% direction 不是camera 看向的方向，而是相反的。

% 我們這里和上面的帖子不同，direction 還保持camera 看向的方向

% 也就是 (right, direction, camera_up) 是一個(gè)右手系

% https://learnopengl.com/Getting-started/Camera 這篇文章中的

% 的lookat 的矩陣也是怪怪的，可能也會(huì)導(dǎo)致模仿者出錯(cuò)。

function ca_axes = camera_axes(up, direction)

???up = up(:);?% 確保得到的up是一個(gè)列向量

???direction = direction(:);?%確保得到的direction 是一個(gè)列向量

???%確保輸入的direction是一個(gè)單位向量

???norm_d = norm(direction, 2);

???if(norm_d < eps)

??????error('輸入的direction向量接近于0');

???end

???direction = direction /norm_d;

???camera_right =?cross( direction, up ); %?drection 叉乘 up

???% 這個(gè)地方要檢查一下 是不是direction 和 up 貢獻(xiàn)

???norm_r = norm(camera_right,2) ; %計(jì)算camera_right的2范數(shù)

???if(norm_r < eps)

??????error("up 和 direction 共線")

???end

???camera_right = camera_right / norm_r; %把camera_right單位化

???camera_up = cross(camera_right, direction); %?right 叉乘 direction

???camera_up = camera_up / norm(camera_up, 2);

???ca_axes = [ camera_right, direction, camera_up];?

end

%

% gluLookAt 矩陣

% 要求相機(jī)位置ca_positin, 目標(biāo)點(diǎn)target, 上方向up 都是長度為3的向量

function?Mat =?gluLookAt(ca_position, target, up)

???ca_position = ca_position(:)

???target = target(:)

???direction = target - ca_position

???disp('在gluLookAt 計(jì)算的direction')

???direction = direction / norm(direction, 2);?

???right = cross(direction, up);

???%注意檢查 up 和 target-ca_position 是否共線??

???norm_r = norm(right,2);

???if(norm_r < eps)

??????error('direction 和 up 幾乎共線, 請(qǐng)修改up方向');

???end

???right = right / norm_r;

???ca_up = cross(right, direction);

???ca_axes = [ right(:), direction(:), ca_up(:) ] % 分別對(duì)應(yīng) \xi , \eta, \zeta 三個(gè)軸

???%確定了right, direction, up 方向之后, 我們確定gluLookAt矩陣

?平移_mat = [ eye(3), -ca_position; [0 0 0 1] ]

?旋轉(zhuǎn)_mat = [ ca_axes', zeros(3,1); [0 0 0 1] ]

???Mat =?旋轉(zhuǎn)_mat * 平移_mat;?

end

%繞著方向 n 的旋轉(zhuǎn)變換, 逆時(shí)針旋轉(zhuǎn)theta角度

function Mat = glRotate(theta, n)

???n = n(:);

???n = n / norm(n, 2);

???n1 = n(1); n2 = n(2); n3 = n(3);

???theta = deg2rad(theta);?% 角度轉(zhuǎn)成弧度

???c = cos(theta);

???s = sin(theta);

???Mat = [ ...

??????c + n1^2*(1-c),??????n1*n2*(1-c) - n3*s,?n1*n3*(1-c) + n2*s ;

??????n1*n2*(1-c) + n3*s,??c + n2^2*(1-c),?????n2*n3*(1-c) - n1*s ;

??????n1*n3*(1-c) - n2*s,??n2*n3*(1-c) + n1*s,?c + n3^2*(1-c);

??????];

end