OpenGL中的旋转是可以叠加的?

1. opengl中的旋转

  如:glrogtate(45.0f, 0, 0, 1),是将当前坐标系顺时针旋转45度,然后绘制,

  程序如下:

   int randium = ;
float lineColor[] = {1.0f, 1.0f, 0.0f};
midPoint_1P8Circle(randium, lineColor); //glPushMatrix();
glRotatef(45.0f, , , );//旋转45度
midPoint_1P8Circle(randium, lineColor);
//glPopMatrix(); //glPushMatrix();
glRotatef(90.0f, , , );//旋转45度
midPoint_1P8Circle(randium, lineColor);
//glPopMatrix(); //glPushMatrix();
glRotatef(135.0f, , , );//旋转45度
midPoint_1P8Circle(randium, lineColor);
//glPopMatrix();

  程序分析:

  (1)因为没有添加glPushMatrix(), 和 glPopMatrix(),所以每一次旋转都会产生相应的旋转矩阵。从而导致最终的旋转结果进行叠加处理。

  (2)glPushMatrix(), 和 glPopMatrix(),所的作用是,保存当前坐标系,并重新建立一个初始模型视图MoldView的变换矩阵,后面的旋转不影响前面的的绘制。

  (3)每一次的旋转,在同一个坐标系下面,即在同一层此上的glPushMatrix(), 和 glPopMatrix(),中间的所有模型处理,旋转只影响该函数后面的模型绘制。

  通过以上分析,即可以解释产下属现象的原因,从本质上来讲,产生这种现象也是必然的。如果相应的添加glPushMatrix(), 和 glPopMatrix(),取消前面的注释,通过1/8最小化的分圆部分,绘制出如下的旋转圆,如右图所示。

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" 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HJcBjfAj3e3WlzCyNkootIHP337QXSaGY4jG+BHm8f3R111zT8Q1w+xbToIuKjf3ra8Prd3O++IY2VDofxLdDj7aO7vCUVMtFFpI1+nnffkMbqh8P4Fujx9tHdX+/dNr4ZQBxdROLon5423qw/yH8j48cb11korprhML4Ferx9dFfPW1K5joppRbrIl8tfkedGbzgI35Ux8b5G0ljNLml8C/R4++jux2+vGd8MII4u8hV8xZ8eJ//7QvhWxd2dzfjC4qSx4l3S+Bbo4V+OUPpPcKaIYugiX8FX/Omx+2Izcv9+bzj469trpFHhLml8C/Twnj1V3UWgmAZHVJjeQWeRNCriyg7pd5FbGCEQXUQx2S83frrc4jONi3Flh9TQRa6johi6iDQTZ5XslxtJYzGu7JB+F7mFEQLRRZTR97YzXm4kjQW4skPSRYhFF5Ft4lljbzjIeLmRNE7LlR3S76KKWxi5fIqS6CIqsfd4JSONXr9rfAtt4dAOqaiLQEl0ERPlPP72+t2Mlxv1rBFdAw7tkP5Bk/HNACLoIiqU/XIjaczDoR3Sf1pU/mO5jooyiCLyyznbZL/cSBoncmif9J8Tuy94/RmC0EUokvFyI2nM5tA+6T8hKryFkTNFlEcXoU7Gy4176yvGN08sh/ZJDpQgEF2EUhkvN/KRjWkc2ic/3rjOLYyQhi4ip8IXqDJebiSNiRzaJw/nb540m+/uVPY84DoqyqOL0CPt5UbSGOfQPul3kVsYIQpdxFTKHI6nvdxIGiMc2if9Lpb/dGJOE1Ehugid0l5uPOi0+VyqgEP7pN9FFbcwAoXRRWjWGw4+LM7H08hHNgYc2ieDLjL2kIMuooDyV632V5dJYxqH9sl3d9qV3MLIdVRUiC7ClF+2foy/3EgaN5zq4ttHd/2Bf721bnxjAB9dRDGVHKDv7mzGX24kjQ7tk0EXuYURctBFmJX4cuPHG9dd/shGh/bJ8l3kCioqRxchQfzlRpc/zdihfTLoIrcwQg66CCHiLzc6m0aH9snyXeR8EZWjiyhGxXQUf7nRzTQ6tE++WX8QDLbxjQF8dBGixF9udDCNDu2T3rOndBGiEEWUoe4KVuTlxk+XW79s/Wj899XGod0y3MVin07MdVRUiy5CrPjLje58SJ9Du2W4ixV+OjFQGF1EeeqO1+MvNzqSRod2y3AX36w/ML49AF2EcPGXG11Io0O7ZbiL097CyBVUqEAXYYXf7t12Ko0O7ZbhLr5vLxjfHoAuwhaj7lr45cb91WXjm6SOW7tlMKjT3sLI+SJUoIsoSefUtLuz+ffMVRc+zdit3TIY0eNvrhrfGIAuwi694eBobrb2aXRrtwxfIje+MQBdRCU0X9AKv9xYyzS6tVuGu/j81Xb+/5HrqFCBLsJS4Zcb65dGt3bLcBe5hRHG0UVURf+xe/jlxqO52Tp9ZKNbu2W4i3w6MYyji7Ba+OXGOn2asVu75ccb16e6hZHLp1CKLqIGgpcba5NGt3bLw/mbQRd/vXfb+PbAcXQRVTF7EB+83FiPNLq1W4a7mOcWRs4XoRRdRG0ELzfWII1u7ZbTdhFQii6iQsaP44OXG49nrlj9kY1u7ZbhLnILI8wiiqgl/+VGqz/N2K09s0AXjR+Coa7oIupq1F2zOo1u7Znv7rS5hRFC0EXU2O7O5ufWJUvT6Nae+fbRXboIIegiqiXt4lZvOPjr22s2plHW46hapIvcwgiD6CJccNBZ/HS5ZddHNrq1ZxboIqAIXUTlZB7H+y83WpRGiQ+iOpEuZnw6scynF+qELsId/suNtqTRrT0z0kVuYYRBdBFO8V9utCKNbu2Zb9YfhLv46XLL+CbBWXQRDjroLMpPo1t7pvfsabiLGbcwch0VqtFFVI6JqxJuPYjxLk716cRAhegiIJNbe2a8i2m3MHLYBdXoIlRg7irPrUcw3kX5V7pRV3QRkMmtPTPeRW5hhCl0EZDJrT0z3sV3d9pp38zlCChFFwGZnNszI13kFkaYQhdROY7mK+Hcgxjp4sdvr0W+gScW9KCLgEzO7ZmRLvLpxDCCKEIdDu5Lcu7hi3exNxwY3yq4hi4CYjm3c8a7GLmFkUMtaEAXAbGc2zkndhHQgC4CYjm3c368cZ1bGGEcXYQiXPEqz7lH8HD+ZnYXeVZBA7oIiOXczhnvIrcwQj+6CKU4vi/DuccuTxd5SkE1ugiI5dzOGe8in04M/egiIJZzO2e8i9zaD/3oItThildJzj187+60413cfbEZ/h6eVVCNLgJiObdzvn10l1sYYRxdhGoc3xfm3AOX2MU36w/C38PzCarRRUAs53bOxC5yaz80o4uAWM7tnIldzPh0YkAFugiI5dzOmdjFyC2MXEeFanQRSjGJleHcY/dm/UG8i8ffXDW+YXAKXQTEcm7n9J49jXfRv4WRIyzoQRQByZzbP9O6yKcTQxu6CEjm3P6Z1kVuYYQ2dBGqcfWrDOceu7Quvt5a3+DJBC3oIiCZc/tnWhe5hRHa0EVowFF+Yc49cHQRxtFFQDIX98/ELvq3MHKEBQ3oIiCZi/tnRhcBDegiNOAovzAXH7jELp40m31v+z//z/846Cy+by8Y30jUGF0EJHNl/+x726Pu2kFn8XjmSloXA6PumvENRo3RRUAyV/bPw/nZiTnkHn/oQRcByVzZP5+/2s4ZRT5bA6rRRajGi4tlOPTYvbvTztNFr981vqmoN7oISObQ/tkbDj63LmVH8Z+z/2p8O1F7dBEacMpYmFsP3NvV5ewuvll/YHwjUXt0EZDMuf0z+/2oz19tG99C1B5dBCRzbv8cddfSovi+vcCVB2hAFwHJXNw/0+7Z8D9SA1CKKEIPjvILc/GB8/rdeBSPZ65s8EyCenQREM7RXTR+z8av9277f0UaoRRdBIRzdBeN3+bPO26gB12EBhzfl+HuYxe+Z+Nwftb/Q55MUI0uAsK5u4uGb/NnoXBoQxcB4ZzeRf17Nj63LoX/kFNGKEUXAeFc30WPZ66wUDh0oovQgOP7Mlx/7Lx+d/fFpvHNgDvoIiAcu2gUx1lQii5CD6aywnjgAK3oIiAcuyigFV0EhGMXBbSii9CD66iF8cBF8WSCUnQREI5dFNCKLgLCsYsCWtFFQDh20Siuo0KdRuP0tNFIZHzbUCfMY2Xw2AH6NBqnaX+V1ssMxn8doJbYtQB9MrpYACkFVGA/AfSptosF0FEXcBG1JB6+KJ5SUMd4F6fFKSkcxJMY0Me6LhZASmE7npGAPi50sQA6ClF4hiXgUioUoYuV4JR0wuPDDFYODx+gD100hZQiP8Y+FcdcqBxdtAgddRZjCehDF2tMyCkpB/Tl8QgC+tBFhAlJKSJ4lFNx2IXK0UWUREc14FED9KGL0Mbrd71+l1PSAngIAH3oIpQKX+U6nL95OH+zyA9xPqV1+30AsYgiNPDT+PzV9kmzedJsev2ujn+0Xu20aVs14/VFVIsuQpt3d9p+F4udMjqOqR/QhC5CtcjJos5Txjqhi4AmdBF6BCeLnDIWQxdTcR0V1aKL0KA3HISjyCljAUz9gCZ0Eaqdnjberi7Hu8gp41To4gScNaIqdBGq9YaDT5db8S5yyjgVJn1AE7oIpdJOFjllnBZdBDShi1At7WSRU8ap0EVAE7oIpUbdtYwocsqYH12cgNcXURW6CKX+efNfs7vIKWNOTPqAJnQR6kw8WeSUMT+6CGhCF6FOnpNFThlzoosTcB0VVaGLUMTrd3NGkVPGPJj0AR2IItQ5nL85MYfGN9IidDEXzhpREl2EHnSxPKZ7QAe6CD0+ty7RxZLoIqADXYRSwTWttGuqxrfQInQxF66joiS6CD0Su/i5dcn4hlmE6R7QgS5Chfghe2IXeQ/qVOgioANdhB6RDyWmiwXQRUAHugg93j66SxdLoou58PoiSqKLqFzivEQXy2O6nwJ1RGF0EXrQxfKY6AEd6CIql3ik/nprnS6WRBcBHegi9PCePaWLJdFFQAe6iMolni/SxfLo4hR4fRGF0UXoQRfLY6IHlCOK0CnexbeP7hrfKovQxalx1ohp0UVULmMiooslMcUDytFF6EQXS6KLgHJ0EZXLOF+Mf9QUXZwKXZwa11ExLboIneJLh9PFqTDFA8rRRehEF0uiiwVx1oj86CKqlT3/0MWSmNwB5egidPr13m26WAZdBJSji6hEzstU8aXD6eJU6GJBXEdFfnQROsW7OOquGd8qizC5A8rRRVSi8Pmi9+yp8Y23CF0shbNG5EEXoVN8iVS6OBWmdUA5ugid6GJJdBFQji6iEjkvUNHFkuhiKVxHxUREEZrtvtiki2UwrQNq0UVUYqqjcLpYBl2sAGeNyEAXoR9dLIMJHVCLLqKMYofddLEM6V08mpvte9vGNwMojC5Cv+OZK+Eu7r7YNL5JFpHexV+2fvzcurS/utwbDoxvTBquoyIDXYR+kaXDjW+PXSyY0Pve9tHc7PHMlV+2fjS+McC06CLKKHbYTRfLsKCLvv3V5ZNmU/JlVc4akYguQj+6WIZNU/nuzuZf3147aTZ/u9eRfFkVCKOLKKPYAXfko6aM/xZ2samLG6OXveHgt3u3T5rNT5dbrBAPK9BF6BdZOtz49tjFsi76vH73c+uSf1l1d0fK+6y4jopEdBGFFZ5V6GIZtk7lveHgw+K8P+QHnUUuq0Isugj93qw/oIuF2dpF397jFf/E8dPl1t76ivHt2eCsETF0EdMqP41Elg43/hvZxfpJvO9t+2/GOWk2P9647vW7xjcJCKOL0I8ulmF9F33+XRxyLqty1ggfUYQR4S5+vHHd+PbYpT7T9+7O5t8zV/3nwafLrf3VZeObBNBFFFD+wPr5q+2gi4fzN43/RnapTxc3Ri97w8FBZzF4NhzPXOGyKsyiizCFLhZWqy76/CVVg+fEh/aC/iVyuI4KH11EAZVMIHSxsHpO373h4GhuNnhacFkVptBFmEIXC6tnF317j1fC78jSv/I4Z42gi8iv2hnj443rdLGYmk/cwZKqAc0rj5NGx9FFmBIsHU4Xp+XErP3b+SV0T5pN4R/oiNqgizCFLhbmRBc3Ri+9fje4iyO4rMrK41CNLiKnyq8t0cXCXOnixvklVcOXVZWuPM51VMfRRZgSLB3+7k7b+MbYxblZe9RdC9/F4eMDHaEIXUROlR9DB118++iu8d/OLs51cWP0su9th+/iCO7lUHdZlbNGZ9FFmEIXC3N3vg4vqRq+rMoSOagQXUQ2dQfNwUdN0cVpudvFjaS7OJSuPM5Zo4PoIkwJlg6ni9NyfabuDQfxuzhEfaAj7EUUMZG6w2W6WJjrXfRFllQNfz5L5ZdVOWt0B12EQXSxMOboM4l3cQSXVfWvPI4aoItIo+H4uDcc0MVi6OI5e49XEk8cWXkcBdBFmEUXi6GLUWlvxqnqAx25juoOugiz6GIxzNHJEu/iqPADHamjC+gi0uiZAfwpy3v21Pjvaxdm51TxJVVZeRxToYswy18ilS5Oiy5m6Q0HB53FtDSW/EBHThlrjy4iQvNeTxeLYWqeLO0ujmCJHN6tikR0EWbRxWLoYi6JS6pWclmVs8Yao4sI07+zv28v0MUCmJSnsPd4JSONSlceh43oIszylw6ni9Oii9PJuIsjuKyq9AMdYRG6CJ+py0J0sRi6WETikqph+T/QkeuoNUYXYRZdLIZJuSCv3814Mw6XVbFBFzF6uWH02HfUXaOLBdDF4jKWVJ125XHOGuuHKMI4f+lw7rSeFtNxWaPuWvaJ40nuD3SkjnVCF2Gc30Xjm2EdJuIK9L3t7DfjnPCBju6hi46TcJhLF4sxP3K1kbGkaniJnMTLql6/6y+dI2FfQiXoIozzP2rK+GZYh1m4Srs7mxlLqgbiK48fzt88/uYqLwPUCV10nJBjXLpYgIiRq5PecDDxLo6T8x/o6PW7wd0d/p8I2aNQBl0ELMX8q0T2kqqRy6r+Goa+8JoA1NFqdNFN7LY1wBCq0hsOspdUTehysfMAAAeLSURBVLuvw/iWoxJ0EbAUXVRr7/FKnhPHsMjbVjn8tBRddBM7bA0whMpNXFI1/tLj81d8cJX16CJgKbqoSZ67OALv2wvGNxgl0UWncJpYJ4ylPl6/m+cuDp9/O+MG+5u16CJgKeZcrfIsqXr2VlVuZ7QcXXQBh621xKDqFtytOBG3M1qNLgKWYsLVLXy34kTczmgvughYiqlWq/wni9zOaDWi6AiOVmuJQdVqqpNFbme0F10E7MUkq9vuzqb37Omou7b/w3f7P3z359zsn3OzGff+czujjehivXF4Wm+MriC94cB79tR79tRP5h+d9p9zs//xj3+L3M7IPikfXQTsxQxrK+ooGV2sK/Y7FzDGQPXoImAvumg3jl5loou1xO7mCIYZqB5dBOxFF+uAw1hp6GLNsIs5hcGuCfZbUegiYC8m01qhjkLQxXpgh3ITow5Ujy4C9qKLQPXootU4TXQcw19D7NXG0UXAXkygQMWIou04snQcw19b7Num0EXAakydQMXooqU4lISP50H9sbdrRhcBqzFjAhWji9bh2BFhPBtcwZ6vDV0ErMZcCVSMLtoifLDo9bsHnbbxTYIEdBGoGF20y6i7djxz5aTZ/PXebeMbAwnooluCA2Quq6pDF4ULnvxBEX2vt9aNbxskYHIEKkYXhesNB/ury58ut4Ii+nrDgfFtgwR00WmcNapAF2U6PW2kFfGk2Tycv2l8CyEE0yJQMboo0H/9v/920FmM5zDwZv2B8Y2EEHQRLzc4cawUXRSl722fdC9kFNG3+2LT+KZCCGZDoGJ0UQiv3z2am51YxJNm83PrkvGthRx0EagSUZRj1F3LE8WTZvPdHe5cxBhdxJh/NZVrqmXQRVFypnHUXTO+qZCDGRCoEl0UInyT4sQucocGwugiUnHiWABdNC7yvJ3YxY83rhvfZojCxAdUiS6KsruzmXi3YhjLvyGCLmICXnScCl00KPIs7Q0Hx99cnXgR1Xv21PiWQxQmO6BKdNGIxOO29+0F7tBAAXQRqBJd1CztSsbe+kqed6K+by8Y/xUgDV3EdLigmo0uSuD1uxktPOgs/rL14+fWJZZ/QyLmOKBKdFG1iZ+V1hsOMt5rc9BZDL7tw+L881fbxn8jSEMXUQRvxklDF9XJ+Xw7nL85MYpABuY1oEp0UYX8R2BvV5fTovjXt9eM/yKwAl1EWROvazmFLhr0y9aPGVFkURvkxEQGVIkuViv/wdbzV9tpLysSRUyFLgKVIYrVmuoKxMcb19PuUCSKmApdRMVcfksOXazKtM+fg047LYq7O3zgMKbj4uQFKEIXjUhbGZwoohi6CLWcOn2ki4UVfvdWxsrgRBHFODFbAXrQxQLKHDP1hoO0lxX5qGEURhehT+3PGuniVMo/H9JWBieKKKPm8xRkqmsg6aJOaSuDE0WUVM/pCTCCLmarcAmI3Z1NVnqDInQRxtTvLTl0MU21o5y2MjhRRCXqMyUBxtHFOBXHPYkrgxNFVIUuQoTIFTZLTyLpogaJK4OzJjgqZOXsA8hEFzcUryOf+IHDLH+KatFFoDKOd1H1WX7iyuBEEZWji5DLuourrnUxPCIaRid+Cz9rgkMF6RMNYAuXo6jBb/c6LH8KPegiLBM5d5RzEulaF3WKrwxOFKGOlDkFKENCHevdRc2XTMMSVwYnilDH/GwC1EPNumgwhGGJK4Oz0huUoouoj8T36Wib023vopAQRsQ/cJgoQjUpz37AdtZ1UWYIw+IrgxNFaCBxZwAqV/h9Okdzs16/m+c75XdRfgjD4iuDs9Ib9JC+bwAqJL6pNfG6qz8j56+jKPLjl6Y3HBx/c5UowghbdxtAj/DULLaOefpnVyMjK4MTRehk064CaBNUJL4ap8E6TrwabNeV0jSRlcFZExya2brnoDa8fje/vfWV/dXlnD60F47mZnM6nrmS+Dm3aY7mZv/zj3/ZCF19jfzHtBKTNvFH2Ru/jOcDy5/CrLrtVC7re9v5A/N6az1/YH7/fil/YI7mZqcKjNUmnjvGC5fYPAf7lyjygcNEEUY4sbPVj/EeIEzs647WCd/Cz5rgMIUu2s179jRi/4fvIv6cm40wHpJaoo4lhVcGZ/lTGEQX8XKDvtJF037Z+pEoQgi6iCq52dc/Ou2+t238wbdXZGVwogiz6CIksqWvFLG8yMrgrPQG4+gi6kxRXz+3Lu2vLvOukEqEVwYnipCALgKpKKJq4Q8cJooQgi4CqcJF/HvmKhN3tcIrg7PSG+Sgi0AqiqhOeGVwoghR6CKQilsv1HnfXiCKkIkuAtAtWBmcNcEhEF0EoFWwMjjLn0ImughAn2BlcKIIsegiAH38DxxmTXBIRhcBaOKvDM7ypxCOLgLQwV8ZnChCProIQLnnr7b9lxWJIuSjiwDUClYGZ3kEWIEuAlDLXxmcKMIWdBGAWqPuGlGERegiAABjdBEAgDG6CADAGF0EAGCMLgIAMEYXAQAYo4sAAIzRRQAAxugiAABjdBEAgDG6CADAGF0EAGCMLgIAMEYXAQAYo4sAAIzRRQAAxugiAABjdBEAgDG6CADAGF0EAGDs/wO5bZw29idovwAAAABJRU5ErkJggg==" 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