cse 167 (fa21) computer graphics: projective geometry

Albert Chern

CSE 167 (FA21)Computer Graphics:Projective Geometry

Perspective• Foreshortening

2

‣ Further objects look smaller

‣ The scaling is non-uniform

Perspective• Orthographic view

3

‣ A view preserving parallelism, proportions‣ Telescopic view standing infinitely far away.

• Perspective view

‣ Does not preserve parallelism, proportions‣ The vantage point is immersed in the scene.

‣ Lack a sense of depth ‣ Provides a sense of depth

GoalMap points from the camera coordinate to the Normalized Device Coordinate (NDC)

• The viewing frustum is mapped to the viewing box in NDC;• The map is a collineation.

Viewing frustum

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Viewing box

so that

FrustumIn geometry, a frustum is the portion of a pyramid (or cone) that lies between two parallel planes cutting it.

Viewing frustumIn graphics, a viewing frustum is the 3 dimensional region that is visible on the screen.

camera

coordinate

system

Viewing frustumIn graphics, a viewing frustum is the 3 dimensional region that is visible on the screen.

• Parameters for a viewing frustum:

camera

coordinate

system

‣ field of view: angle �<latexit sha1_base64="sONfpeQh5hAcRlCpsOoQ/Yy2QjQ=">AAAiGniczZrLbuPKEYZ1Tm4nzm1OgqyyoWM4CRLbEOWxLQMzB2NLsiVfJFkXX2ZkDJqtlkSLt9NNeewhCOQJsk0WeZrsgmyzyTvkIVJNitUkJRkIkEUImGx9VdVd1fzZIikbnmUKv1j81xdffue73/v+D7764dqPfvyTn/7s1dc/vxbujFPWp67l8luDCGaZDuv7pm+xW48zYhsWuzGmFWm/eWRcmK7T8589dm+TsWOOTEp8iQbexPz4aqO4U4w2bbGhzxsb7/7w7186v2nW2x+/Xi8Mhi6d2czxqUWE+KAXPf8+INw3qcXCtcFMMI/QKRmzD9B0iM3EfRAlG2qbQIbayOXw5/haRNMRAbGFeLYN8LSJPxF5m4TLbB9m/qh8H5iON/OZQ+OBRjNL811NVq4NTc6obz1Dg1BuQq4anRBOqA/zk+lJjAQlFhu+9fmMbUHzLbVhvyXdwfk+kIMPmTDHTrbYOOw+mDDrkfkZW/A0gqG2HJMy2cjaCKUwlyIL5SC+61oSY1swP5C+jMuJyw4fqM4301iYM8f0n7K+EkHPm5rwuTll/oS7s/FEswmdgJT4c6Yux+U2VGbfBzPYZzuixKHMCtfWNje1br121NGOW7e17trAYZ8EeWSG+xQMRq4LxzCC1LVt4gyDwaOYMMIl/1C6/7Czd7+macEm7DRtYAOOWogiLNK9BRulMGWbiKnpDT4N52Zl8IYjmUgWMB/mlJtPga5t6BroPN2VZREvkEWmh0s7QA+cCd/lSa+RMVff5P+kvqKs8H9TH5zjWrOaOc/A5KZVW/3ji5p23DmqnNd6XanZKSOwIPkg1rVBlcFSwdkJXPMnxDat56B1eRsGl04XrnXXqoVBiE4xkq5gvyB8PCciXAiyw0BeBF3mvxAE++FiqBGHphLrTojHlg8RRPPwZnv/G01D256Eb/a3D9JwP4IH2+U0PIhgefswDcsRPNzWi98oeAjwjV7c1kspqBcjWtqG8BQtraVnbXUFxsoKto9hapaWEVmW1hJZlhYUWZZWFVmWlhZZltcXm+ZFgo6kbJ2xxQbvZs6QjWCZGsac80U+n5RLWDir8O1om6DCIOkhDKIV1fWYk1J9ZlsQ0W/1/dfhcrpqtHle89Go5Qr2Xwx3oC8bDmhyabn+BK6s+eUnL8vWycJVuLb5dnGDVf+k1ezBZdzq9OpHzWp3udsAZnNwdBSnbxjBURjG7PgY2XHCKhVklYRVq8iqCavVkNUSdnKC7CRhp6fIThNWryOrJ6zRQNZI2NkZsrOEnZ8jO0/YxQWyi4RdXiK7TFiziayZsFYLWSth7TaydsKurpBdJazTQdZJWLeLrJuwXg9ZL2H9PrJ+wq6vkV0n7OYG2U3Cbm+R3Sbs7g7ZXcLev0f2HlgMjUQIIyUE4xgZCsGoIEMhGFVkKASjhgyFYJwgQyEYp8hQCEYdGQrBaCBDIRhnyFAIxjkyFIJxgQyFYFwiQyEYTWQoBKOFDIVgtJGhEIwrZCgEo4MMhWB0kaEQjB4yFILRR4ZCMK6RoRCMG2QoBOMWGQrBuEOGQjDeI1NCoHMhwM2uUgI9VhClQCsKohZoVUEUA60piGqgJwqiHOipgqgHWlcQBUEbCqIi6JmCKAl6riBqgl4oiKKglwqiKmhTQZQFbSmIuqBtBVEY9EpBVAbtKIjSoF0FURu0pyCKg/YVRHXQawVRHvRGQdQHvVUQBULvFESF0PcKKomIuUQE5Uoi4lhBlIioKIgSEVUFUSKipiBKRJwoiBIRpwqiRERdQZSIaCiIEhFnCqJExLmCKBFxoSBKRFwqiBIRTQVRIqKlIEpEtBVEiYgrBVEioqMgSkR0FUSJiJ6CKBHRVxAlIq4VRImIGwVRIuJWQZSIuFMQJSLeK5j6OiG4tBBcbgxkBjKKjCIbIhsiY8gYstEI4QjhGNkY2QTZBJmJzET2gOwB2RTZFJmFzEJmI7OROcgcZC4yF5mHzEP2LbJvkXFkHJlAJpD5yHxkM2QzZI/IHpF9QvYJ2ROyJ2TPyJ6RfUb2WSnhM+Oq5qLky25Ho/vWq/5Rr9ZpNlrN5fes8q419XLCMOwwvkOWt8xCI5rwny3zMxtq+mayLXvuNOJnxn6YtOzM06LyPLK8CTFYMp+xc3zIvg1wneRBILIGOjxwJ1ucnEoNCs3EmiaG2poZ5s0PDynzw4J5Ok2Zp7m04q6njDv6zp49SyVohimei4pHXBb18EJUnMiyqGkmSlvbXLK9oIp2p3VWq/Qa1zWt2z6q1FZrI51Op43zEnSipTZtraSslQVrPWWtt1+U7OVRr6612rXOUa/VWZkZOHJGrC3NtMnYdAh/3tJgJC16yNM8wn2huSPNmdlG9LqSs3Q6DZDnAB5nOTwVcvm+NWhEa0zWq8PyXh2WL0yMnbyTRGHmGbc1NwcD4hPwl/uSnANNvocmXCPWmBmcxNe3z7Vcjz4uTkN55WRsQBJjLbpu0kYgibHu2jkjkJV5XhIPnqZhFwebdj4lE9fkqXx0z9iA4HeZRxyxkLOkiYeZz9nElMnwIWcDgjdS7ihnBBIJS3M937TNz9Er+6X1/R5OBB/bpjwV8fElN/IUu5Gnl9xMZxS5yeMLbmLmRW7yKHO9MJk25u7ME5F85ce5GMSKU3N6EQbwt8LaBWt3pbUtze2X7H1p76/svQW9t1ZaIVjGbi43C3e+Bow4mQbwKdRg5RUuFAyX8vBZG8Yvo3ZWdC9mmfjZSvGOrbTj2FrpKCyLZPpc7Vo1R3B+o/2q6mOX7os+ntRctF+TZ7/LqAtnfTRzhkT+PEQs+UuPvSpd2X9jsN6IlHPNKFCtQiw6s2YiuyyNORniW7v4yO0gouHAcaNXbmJhKVsaJF6OojNuLQZFdGUMd/3FEAkzEdnFeGg+LsZIuHKUi2ouYP7Yscq/zoRYHCKi6Zhc+RNGp7mJixjMp03EJDf3L8yjDMoWGfeTLTabCnyVc9cghmmZ/nO8Llbcx9y6CCRZNK9Jfq0GEi+atSefcTMjKBnxiQ3HDJZTltwVm3ALLt8b82hZjyyQcuQWJsNEn6LKUkETWPUpC4o7e+wpHHBixj+nbMcg2AoTj21dflZdQnINZ2lyfvo2Ut2ZO04aO+obw/Hlz5fBwOIUyodvqflXkAc3b3GTPeW/q4DEOZwy12a+/KVv6cV5BGtYGET7FR4XzBn7kzCYH1d4XbtwOcndyoVowoacWHIxmrfm5XnpqqkXZ92c2TBzNJkwebp85oGoqszyiemHmj9/roClN/pxOTD0MOBjI4RTpe+Vi4eHW8WdXX33oKxDY393/3DvIMxFlDBir7S3u38gHQ/18p5sHB6WD3f3cxFjNUZpb79YLMqI18W4sXcgST4iNcYepLALjuViMc4KAiDTXISnxnitz8fQD0vpwfIRaoz9crFcjpLZ1+PGARRS1nMRXI1RLs7TLyWN+D8B8hHJGMl/CsiuiyUoSWZV3H29nx9jCmPA6vEMg+QtJbTsHGTWk0HF+KDfwxISn58NPcyZxynzeNHspczeopmnzHzRPE2Zp9L88dWGnv8PicXGdWlH390pXekb767+WIi2rwq/Kvy68LuCXjgovCvUC+1Cv0ALk8KfCn8u/GX9r+t/W//7+j9i1y+/iI+FXxQy2/o//wObtlNl</latexit>

‣ aspect ratio: a = widthheight

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‣ near clipping distance n> 0<latexit sha1_base64="ldDKvRcqcTDRwCxO41iTnChPc3c=">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</latexit>

‣ far clipping distance f ( f > n)<latexit sha1_base64="7wRXczYZShMYm0+D/vBdk3kjya4=">AAAiHniczZpZb+PIEce1m2vjXLPJY17oGEY2gW2I8tiWgZnB2JJsyZdkHb5GxqDZakm0eG035bGHIJDPkLccD/k0eQvymv02qSbFapKSDCyQhxAw2fpVVXdV888WSdnwLFP4xeJ3X3z5gx/+6Mc/+eqnKz/7+S9++atXX//6SrhTTlmPupbLbwwimGU6rOebvsVuPM6IbVjs2phUpP36kXFhuk7Xf/bYvU1Gjjk0KfEB3Qz7G98M3zl/+PhqrbhVjDZtvqHPGmvv9e/+AttfWx+/Xi30By6d2szxqUWE+KAXPf8+INw3qcXClf5UMI/QCRmxD9B0iM3EfRAlHGrrQAba0OXw5/haRNMRAbGFeLYN8LSJPxZ5m4SLbB+m/rB8H5iON/WZQ+OBhlNL811NVq8NTM6obz1Dg1BuQq4aHRNOqA9zlOlJDAUlFhu89fmUbUDzLbVhvyHdwfk+kIMPmDBHTrbYOOw+GDPrkfkZW/A0hKE2HJMy2cjaCKUwlyIL5SC+61oSY1swP5C+jMuJyw4fqM7X01iYU8f0n7K+EkHP65rwuTlh/pi709FYswkdg5z4c6Yux+U2VGbfB1PYZzuixKHMCldW1te1Tr120NYOmze1zkrfYZ8EeWSG+xT0h64LxzCC1LVt4gyC/qMYM8Il/1C6/7C1c7+iacE67DStbwOOWogiLNK9BWulMGUbi4np9T8NZmZl8AZDmUgWMB/mlJtPga6t6RroPN2VZREvkEWmh0s7QA+cCd/lSa+RMVff+P+kvqKs8H9TH5zj2kU1c56ByU2rNnuHZzXtsH1QOa11O1KzE0ZgUfJBrCv9KoOlgrMjuOaPiG1az0Hz/CYMzp0OXOuuVQuDEJ1iJF3Bfkb4aEZEOBdkh4G8CDrMfyEI9oP5UCMOTSXWGROPLR4iiObhzebuO01D246Eb3Y399JwN4J7m+U03ItgeXM/DcsR3N/Ui+8U3Af4Ri9u6qUU1IsRLW1CeIqWVtKztrwCY2kFm4cwNQvLiCwLa4ksCwuKLAuriiwLS4ssi+uLTbMiQUdSts7IYv33U2fAhrBMDWLO+TyfTco5LJxV+Ia0TVBhkPQQBtGK6nrMSak+s82J6Pf67utwMV022iyv2WjUcgX7HsPt6YuGA5pcWq4/hitrdvnJy7J5NHcVrqy/nd9g1T9qXnThMm62u/WDi2pnsVsfZrN/cBCnbxjBQRjG7PAQ2WHCKhVklYRVq8iqCavVkNUSdnSE7Chhx8fIjhNWryOrJ6zRQNZI2MkJspOEnZ4iO03Y2Rmys4SdnyM7T9jFBbKLhDWbyJoJa7WQtRJ2eYnsMmHtNrJ2wjodZJ2EdbvIugnr9ZD1EnZ1hewqYdfXyK4TdnOD7CZht7fIbhN2d4fsDlgMjUQIQyUE4xAZCsGoIEMhGFVkKASjhgyFYBwhQyEYx8hQCEYdGQrBaCBDIRgnyFAIxikyFIJxhgyFYJwjQyEYF8hQCEYTGQrBaCFDIRiXyFAIRhsZCsHoIEMhGF1kKASjhwyFYFwhQyEY18hQCMYNMhSCcYsMhWDcIVNCoDMhwM2uUgI9VBClQCsKohZoVUEUA60piGqgRwqiHOixgqgHWlcQBUEbCqIi6ImCKAl6qiBqgp4piKKg5wqiKuiFgigL2lQQdUFbCqIw6KWCqAzaVhClQTsKojZoV0EUB+0piOqgVwqiPOi1gqgPeqMgCoTeKogKoXcKKomImUQE5Uoi4lBBlIioKIgSEVUFUSKipiBKRBwpiBIRxwqiRERdQZSIaCiIEhEnCqJExKmCKBFxpiBKRJwriBIRFwqiRERTQZSIaCmIEhGXCqJERFtBlIjoKIgSEV0FUSKipyBKRFwpiBIR1wqiRMSNgigRcasgSkTcKZj6OiG4tBBcbgxkBjKKjCIbIBsgY8gYsuEQ4RDhCNkI2RjZGJmJzET2gOwB2QTZBJmFzEJmI7OROcgcZC4yF5mHzEP2LbJvkXFkHJlAJpD5yHxkU2RTZI/IHpF9QvYJ2ROyJ2TPyJ6RfUb2WSnhM+Oq5qLki25Ho/vWy95Bt9a+aDQvFt+zyrvW1MsJw7DD+A5Z3jILjWjCf7bMz2yg6evJtui504ifGXth0rIzT4vK88DyxsRgyXzGzvEh+zbAdZIHgcga6PDAnWxxcio1KDQTa5oYamtmmDc/PKTMD3PmySRlnuTSirueMO7oWzv2NJWgGaZ4LioecVHUwwtRcSKLoiaZKG1lfcH2gipa7eZJrdJtXNW0TuugUluujXQ67RbOS9COltq0tZKyVuas9ZS13npRsucH3brWbNXaB91me2lm4MgZsTY00yYj0yH8eUODkbToIU/zCPeF5g41Z2ob0etKztLpNECefXic5fBUyOX71qARrTFZrzbLe7VZvjAxcvJOEoWZZ9zmzBz0iU/AX+5Lcg40+S6acI1YI2ZwEl/fPtdyPfq4OA3klZOxAUmMtei6SRuBJMa6a+eMQJbmeU48eJqGXRxs2vmUTFyTJ/LRPWMDgt9lHnHEXM6SJh5mPmcTUyaDh5wNCN5IucOcEUgkLM31fNM2P0ev7RfW90c4EXxkm/JUxMeX3MhT7EaeXnIznWHkJo8vuImpF7nJo8z1zGTaiLtTT0TylR9nYhBLTs3xWRjA3xJrB6ydpdaWNLdesvekvbe09yb03lxqhWAZu77YLNzZGjDkZBLAp1CDlVe4UDBcyoNnbRC/jNpa0r2YZuKnS8U7stKOI2upo7AskulzuWvVHML5jfbLqo9dOi/6eFJz0X5Fnv0Ooy6c9eHUGRD58xCx5C899rJ0Zf+N/mojUs4Vo0C1CrHo1JqK7LI04mSAb+3iI7eDiIZ9x41euYm5pWxhkHg5ik65NR8U0aUx3PXnQyTMRGQX44H5OB8j4dJRzqq5gNljxzL/OhNifoiIpmNy5Y8ZneQmLmIwnzYR49zcvzCPMihbZNxPtthsKvBVzl2DGKZl+s/xulhxH3PrIpBk0bwi+bUaSLxo1p58xs2MoGTEJzYYMVhOWXJXbMItuHxvzKNlPbJAypFbmAwTfYoqSwWNYdWnLChu7bCnsM+JGf+cshmDYCNMPDZ1+Vl1Cck1nIXJ+enbSHVn7jhp7KhvDMeXP18GfYtTKB++pWZfQR7cvMVN9pT/rgIS53DMXJv58pe+hRfnAaxhYRDtl3icMWfkj8NgdlzideXC5SR3SxeiMRtwYsnFaNaaleelq6ZenPXF1IaZo8mEydPlMw9EVWWWT0w/1PzZcwUsvdGPy4GhhwEfGSGcKn2nXNzf3yhubevbe2UdGrvbu/s7e2EuooQRO6Wd7d096bivl3dkY3+/vL+9m4sYqTFKO7vFYlFGvC7GjZ09SfIRqTF2IIVtcCwXi3FWEACZ5iI8NcZrfTaGvl9KD5aPUGPslovlcpTMrh439qCQsp6L4GqMcnGWfilpxP8JkI9Ixkj+U0B2XSxBSTKr4vbr3fwYExgDVo9nGCRvKaFlay+znvQrxgf9HpaQ+Pys6WHOPEqZR/NmL2X25s08Zebz5knKPJHmj6/W9Px/SMw3rkpb+vZW6VJfe3/5p0K0fVX4beF3hW8KemGv8L5QL7QKvQItWIU/F/5W+PvqP1b/ufqv1X/Hrl9+ER8LvylkttX//BdCTVXz</latexit>

CollineationA straight line in the physical space should still be a straight line in the viewing box or on the screen.

CollineationA transformation is called a collineation if straight lines are mapped tostraight lines.

• Synonyms

‣ Collineation

‣ Projective transformation

‣ Homography

• Parallelism is generally notpreserved! This is more general than linear and affine transformations.

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vxvyvz

35

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Multiplying 3D vectors by a 3x3 matrixLinear transformation

Linear, affine and projective

24

vxvyvz

35 7!

24

a11 a12 a13a21 a22 a23a31 a32 a33

3524

vxvyvz

35


Multiplying 3D vectors by a 3x3 matrixLinear transformation

Multiplying 3D positions (in 4D hom. coord.)by a 4x4 matrix with 0,0,0,1 as the last row. Affine transformation264

pxpypz1

375 7!

264

a11 a12 a13 bxa21 a22 a23 bya31 a32 a33 bz0 0 0 1

375

264

pxpypz1

375

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264

pxpypz1

375 7!

264


375

264

pxpypz1

375



Multiplying 3D positions (in 4D hom. coord.)by a general 4x4 matrix.264

pxpypz1

375 7!

264

a11 a12 a13 a14a21 a22 a23 a24a31 a32 a33 a34a41 a42 a43 a44

375

264

pxpypz1

375

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Projective transformation(collineation)

24

vxvyvz

35 7!

24

a11 a12 a13a21 a22 a23a31 a32 a33

3524

vxvyvz

35


264

pxpypz1

375 7!

264


375

264

pxpypz1

375



264

pxpypz1

375 7!

264

a11 a12 a13 a14a21 a22 a23 a24a31 a32 a33 a34a41 a42 a43 a44

375

264

pxpypz1

375

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Projective transformationspreserve straight lines.

Affine transformationsare projective transformationsthat preserve parallelism.

Linear transformationsare affine transformationsthat preserve the origin.

Final projection after modelview

model matrix M1

<latexit sha1_base64="lav6xJFFXxxvL/xVy9m/nkUQfWo=">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</latexit>

model matrix M2

<latexit sha1_base64="oNW5p+oF/eUeDREN3KvnYQH7tw0=">AAAiGXiczZpLb+PIEce1m9fGec0mx1zoGEaCwDZEeWzLwMxibEm25JdkvfwYGYNmqyXR4mu7KY89BD9Crskhnya3INec8m1STYrVJCUZWCCHEDDV/lVVd1Xzz+ZDMjzLFH6x+J+vvv7Rj3/y05998/O1X/zyV7/+zZtvf9sX7oxT1qOu5fJbgwhmmQ7r+aZvsVuPM2IbFrsxphVpv3liXJiu0/VfPPZgk7FjjkxKfECdy0+lT282ijvFaNMWG/q8sVGYb61P364XBkOXzmzm+NQiQnzUi57/EBDum9Ri4dpgJphH6JSM2UdoOsRm4iGIcg21TSBDbeRy+HN8LaLpiIDYQrzYBnjaxJ+IvE3CZbaPM39UfghMx5v5zKHxQKOZpfmuJgvXhiZn1LdeoEEoNyFXjU4IJ9SH6cn0JEaCEosN3/t8xrag+Z7asN+S7uD8EMjBh0yYYydbbBz2EEyY9cT8jC14HsFQW45JmWxkbYRSmEuRhXIQ33UtibEtmB9IX8blxGWHD1Tnm2kszJlj+s9ZX4mg501N+NycMn/C3dl4otmETkBJ/CVTl+NyGyqzH4IZ7LMdUeJQZoVra5ubWqdeO2prx83bWmdt4LDPgjwxw30OBiPXhc8wgtS1beIMg8GTmDDCJf9Yevi4s/ewpmnBJuw0bWADjlqIIizSvQUbpTBlm4ip6Q0+D+dmZfCGI5lIFjAf5pSbz4Gubega6DzdlWURL5BFpodLO0APnAnf5UmvkTFX3+T/pL6irPB/Ux8c49pVNXOcgclNqzZ7xxc17bh9VDmvdTtSs1NGYD3yQaxrgyqDpYKzEzjnT4htWi9B8/I2DC6dDpzrrlULgxCdYiRdwX5B+HhORLgQZIeBPAk6zH8lCPbDxVAjDk0l1pkQjy0fIojm4d32/neahrY9Cd/tbx+k4X4ED7bLaXgQwfL2YRqWI3i4rRe/U/AQ4Du9uK2XUlAvRrS0DeEpWlpLz9rqCoyVFWwfw9QsLSOyLK0lsiwtKLIsrSqyLC0tsiyvLzbNiwQdSdk6Y4sNPsycIRvBMjWMOeeLfD4pl7BwVuHiaJugwiDpIQyiFdX1mJNSfWZbENEf9f234XK6arR5XvPRqOUK9gOGO9CXDQc0ObVcfwJn1vz0k6dl82ThLFzbfL+4wap/0rzqwmncbHfrR1fVznK3Aczm4OgoTt8wgqMwjNnxMbLjhFUqyCoJq1aRVRNWqyGrJezkBNlJwk5PkZ0mrF5HVk9Yo4GskbCzM2RnCTs/R3aesIsLZBcJu7xEdpmwqytkVwlrNpE1E9ZqIWsl7Poa2XXC2m1k7YR1Osg6Cet2kXUT1ush6yWs30fWT9jNDbKbhN3eIrtN2N0dsruE3d8juwcWQyMRwkgJwThGhkIwKshQCEYVGQrBqCFDIRgnyFAIxikyFIJRR4ZCMBrIUAjGGTIUgnGODIVgXCBDIRiXyFAIxhUyFILRRIZCMFrIUAjGNTIUgtFGhkIwOshQCEYXGQrB6CFDIRh9ZCgE4wYZCsG4RYZCMO6QoRCMe2RKCHQuBLjZVUqgxwqiFGhFQdQCrSqIYqA1BVEN9ERBlAM9VRD1QOsKoiBoQ0FUBD1TECVBzxVETdALBVEU9FJBVAW9UhBlQZsKoi5oS0EUBr1WEJVB2wqiNGhHQdQG7SqI4qA9BVEdtK8gyoPeKIj6oLcKokDonYKoEHqvoJKImEtEUK4kIo4VRImIioIoEVFVECUiagqiRMSJgigRcaogSkTUFUSJiIaCKBFxpiBKRJwriBIRFwqiRMSlgigRcaUgSkQ0FUSJiJaCKBFxrSBKRLQVRImIjoIoEdFVECUiegqiRERfQZSIuFEQJSJuFUSJiDsFUSLiXsHU5YTg0kJwuTGQGcgoMopsiGyIjCFjyEYjhCOEY2RjZBNkE2QmMhPZI7JHZFNkU2QWMguZjcxG5iBzkLnIXGQeMg/Z98i+R8aRcWQCmUDmI/ORzZDNkD0he0L2GdlnZM/InpG9IHtB9gXZF6WEL4yrmouSL7sdje5br3tH3Vr7qtG8Wn7PKu9aUy8nDMMO4ztkecssNKIJ/8Uyv7Chpm8m27LnTiN+ZuyFScvOPC0qzyPLmxCDJfMZO8cf2bcBrpM8CETWQIcH7mSLk1OpQaGZWNPEUFszw7z58TFlflwwT6cp8zSXVtz1lHFH39mzZ6kEzTDFc1HxiMuiHl+JihNZFjXNRGlrm0u2V1TRajfPapVuo1/TOq2jSm21NtLptFs4L0E7WmrT1krKWlmw1lPWeutVyV4edetas1VrH3Wb7ZWZgSNnxNrSTJuMTYfwly0NRtKihzzNI9wXmjvSnJltRK8rOUun0wB5DuBxlsNTIZfvW4NGtMZkvdos79Vm+cLE2Mk7SRRmnnGbc3MwID4Bf7kvyTnQ5GtowjVijZnBSXx++1zL9ejj4jSUZ07GBiQx1qLzJm0Ekhjrrp0zAlmZ5yXx4GkadnGwaedTMnFNnspH94wNCF7LPOKIhZwlTTzMfM4mpkyGjzkbELyRckc5I5BIWJrr+aZtfone2C+t789wIPjYNuWhiD9fcyPPsRt5fs3NdEaRm/x8xU3MvMhNfspcL0ymjbk780QkX/nvXAxixaE5vQgD+Fth7YC1s9LakubWa/aetPdW9t6E3psrrRAsYzeXm4U7XwNGnEwD+C/UYOUVLhQMp/LwRRvGL6N2VnQvZpn42Urxjq2049ha6Sgsi2T6XO1aNUdwfKP9qupjl86rPp7UXLRfk0e/w6gLR300c4ZEfj1ELPlNj70qXdl/Y7DeiJTTZxSoViEWnVkzkV2WxpwM8a1d/MntIKLhwHGjV25iYSlbGiRej6Izbi0GRXRlDHf9xRAJMxHZxXhoPi3GSLhylItqLmD+2LHKv86EWBwioumYXPkTRqe5iYsYzKdNxCQ396/MowzKFhn3ky02mwpcyrlrEMO0TP8lXhcr7lNuXQSSLJp9kl+rgcSLZu3ZZ9zMCEpGfGbDMYPllCV3xSbcgsv3xjxa1iMLpBy5hckw0X9RZamgCaz6lAXFnT32HA44MeOvU7ZjEGyFice2Lv9XXUJyDWdpcn76NlLdmTtOGjvqiuH48uvLYGBxCuXDVWp+CfLg5i1usuf8tQpInMMpc23my2/6lp6cR7CGhUG0X+FxwZyxPwmD+ecKr74Lp5PcrVyIJmzIiSUXo3lrXp6Xrpp6cdZXMxtmjiYTJg+XzzwQVZVZPjH9UPPnzxWw9EZfLgeGHgZ8bIRwqPS9cvHwcKu4s6vvHpR1aOzv7h/uHYS5iBJG7JX2dvcPpOOhXt6TjcPD8uHufi5irMYo7e0Xi0UZ8bYYN/YOJMlHpMbYgxR2wbFcLMZZQQBkmovw1Bhv9fkY+mEpPVg+Qo2xXy6Wy1Ey+3rcOIBCynougqsxysV5+qWkEf8SIB+RjJH8UkB2XSxBSTKr4u7b/fwYUxgDVo8XGCRvKaFl5yCzngwqxkf9AZaQ+Phs6GHOPE6Zx4tmL2X2Fs08ZeaL5mnKPJXmT2829PwvJBYb/dKOvrtTun678aE6//XEN4XfF/5Q+FNBLxwUPhTqhVahV6CFceEvhb8W/rb+9/V/rP9z/V+x69dfzWN+V8hs6//+L0w4T3w=</latexit>

camera matrix C

<latexit sha1_base64="28CNDy9hbIsgNLsEDHazLl6Vobk=">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</latexit>

view matrixV

<latexit sha1_base64="kR5ovrjEGkmlOJPv0jZUf3OcHgA=">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</latexit>

projection matrix

R4NDC

<latexit sha1_base64="voR6u16A2hiugCMKo6+LH0rYx/Y=">AAAiJ3iczZpZb+PIEce1m2vjHDubPOaFjmEkCGxDlC8ZmFmsLcmWfEiyDtszljNotloSLV7bTXnsIQjke+Q1ecinyVuQPOabpJoUq0lKMhAgDyFgqv2rqu7q5p/FQzI8yxR+sfivL778wQ9/9OOffPXTtZ/9/Be//PrNN7+6Ee6MU9anruXyO4MIZpkO6/umb7E7jzNiGxa7NaYVab99YlyYrtPzXzz2YJOxY45MSnxAH998Peh0/rj3MRhwW2tWK+HHNxvFnWK0aYsNfd7YKMy39sdv1guDoUtnNnN8ahEh7vWi5z8EhPsmtVi4NpgJ5hE6JWN2D02H2Ew8BFHmobYJZKiNXA5/jq9FNB0REFuIF9sAT5v4E5G3SbjMdj/zR+WHwHS8mc8cGg80mlma72pyGbShyRn1rRdoEMpNyFWjE8IJ9WGxMj2JkaDEYsN3Pp+xLWi+ozbst6Q7OD8EcvAhE+bYyU42DnsIJsx6Yn7GFjyPYKgtx6RMNrI2QimspchCOYjvupbE2BbMD6Qv43LhssMHqvPNNBbmzDH956yvRNDzpiZ8bk6ZP+HubDzRbEInoCv+kpmX43IbZmY/BDPYZzuixKHMCtfWNje1br123NFOWne17trAYZ8EeWKG+xwMRq4Ln2EEqWvbxBkGgycxYYRLfl96uN/Zf1jTtGATdpo2sAFHLUQRFunego1SmLJNxNT0Bp+Gc7MyeMORTCQLmA9rys3nQNc2dA10nu7KsogXyEmmh0s7QA+cCd/lSa+RMTe/yf/J/Ipyhv+b+cExrjWrmeMMTG5atdU/uaxpJ53jykWt15WanTIC1ckHsa4NqgxKBWencM6fEtu0XoLW1V0YXDldONddqxYGITrFSLqC/ZLw8ZyIcCHIDgN5EnSZ/0oQ7IeLoUYcmkqsOyEeWz5EEK3D2+2DbzUNbfsSvj3YPkzDgwgebpfT8DCC5e2jNCxH8GhbL36r4BHAt3pxWy+loF6MaGkbwlO0tJZetdUzMFbOYPsElmbpNCLL0rlElqUTiixLZxVZlk4tsiyfX2yaTxJ0JGXrjC02+G7mDNkIytQw5pwv8vmiXEHhrMKl0jZBhUHSQxhEFdX1mJNSfWZbENHv9IO9cDldNdo8r/lo1HIF+y+GO9SXDQc0ObVcfwJn1vz0k6dl63ThLFzbfLe4QdU/bTV7cBq3Or36cbPaXe42gNUcHB/H6RtGcByGMTs5QXaSsEoFWSVh1SqyasJqNWS1hJ2eIjtN2NkZsrOE1evI6glrNJA1EnZ+juw8YRcXyC4SdnmJ7DJhV1fIrhLWbCJrJqzVQtZKWLuNrJ2w62tk1wnrdJB1EtbtIusmrNdD1ktYv4+sn7CbG2Q3Cbu9RXabsLs7ZHcJe/8e2fuEffiA7AOwGBqJEEZKCMYJMhSCUUGGQjCqyFAIRg0ZCsE4RYZCMM6QoRCMOjIUgtFAhkIwzpGhEIwLZCgE4xIZCsG4QoZCMJrIUAhGCxkKwWgjQyEY18hQCEYHGQrB6CJDIRg9ZCgEo48MhWDcIEMhGLfIUAjGHTIUgvEeGQrB+IBMCYHOhQA3u0oJ9ERBlAKtKIhaoFUFUQy0piCqgZ4qiHKgZwqiHmhdQRQEbSiIiqDnCqIk6IWCqAl6qSCKgl4piKqgTQVRFrSlIOqCthVEYdBrBVEZtKMgSoN2FURt0J6CKA7aVxDVQW8URHnQWwVRH/ROQRQIfa8gKoR+UFBJRMwlIihXEhEnCqJEREVBlIioKogSETUFUSLiVEGUiDhTECUi6gqiRERDQZSIOFcQJSIuFESJiEsFUSLiSkGUiGgqiBIRLQVRIqKtIEpEXCuIEhEdBVEioqsgSkT0FESJiL6CKBFxoyBKRNwqiBIRdwqiRMR7BVEi4oOCqcsJwdJCsNwYyAxkFBlFNkQ2RMaQMWSjEcIRwjGyMbIJsgkyE5mJ7BHZI7IpsikyC5mFzEZmI3OQOchcZC4yD5mH7Htk3yPjyDgygUwg85H5yGbIZsiekD0h+4TsE7JnZM/IXpC9IPuM7LNSwmfG1ZyLki+7HY3uW6/7x71ap9loNZffs8q71tTLCcOww/gOWd4yC41own+xzM9sqOmbybbsudOInxn7YdKyM0+LyvPY8ibEYMl6xs7xR/ZtgOskDwKRNdDhgTvZ4uRUajDRTKxpYqitmWHe/PiYMj8umKfTlHmaSyvuesq4o+/s27NUgmaY4rmoeMRlUY+vRMWJLIuaZqK0tc0l2yuqaHda57VKr3FT07rt40pttTbS6XTauC5BJyq1aWslZa0sWOspa739qmSvjnt1rdWudY57rc7KzMCRM2JtaaZNxqZD+MuWBiNp0UOe5hHuC80dac7MNqLXlZyl02mAPAfwOMvhqZDL961BI6oxWa8Oy3t1WH5iYuzknSQKM8+4rbk5GBCfgL/cl+QaaPKlNOEascbM4CQ+v32u5Xr0sTgN5ZmTsQFJjLXovEkbgSTGumvnjEBW5nlFPHiahl0cbNr5lEysyVP56J6xAcFrmUccsZCzpImHmc/ZxJTJ8DFnA4I3Uu4oZwQSCUtzPd+0zc/R+/ul8/sDHAg+tk15KOLP19zIc+xGnl9zM51R5CY/X3ETMy9yk58y10uTaWPuzjwRyVf+OxeDWHFozi7DAP5WWLtg7a60tqW5/Zq9L+39lb23oPfWSisEy9jN5WbhzmvAiJNpAP+FGlRe4cKE4VQevmjD+GXUzoruxSwTP1sp3rGVdhxbKx2FZZFMn6tdq+YIjm+0XzX72KX7qo8nNRft1+TR7zLqwlEfzZwhkV8PEUt+02OvSlf23xisNyLl3DAKVKsQi86smciWpTEnQ3xrF39yO4hoOHDc6JWbWChlS4PE61F0xq3FoIiujOGuvxgiYSYiW4yH5tNijIQrR7ms5gLmjx2r/OtMiMUhIpqOyU1/wug0t3ARg/W0iZjk1v6VdZRB2UnG/WQnm00FLuXcNYhhWqb/EtfFivuUq4tAkqJ5Q/K1GkhcNGvPPuNmRlAy4hMbjhmUU5bcFZtwCy7fG/OorEcWSDlyC5Nhov+imaWCJlD1KQuKO/vsORxwYsZfp2zHINgKE49tXf6vuoTkGs7S5Pz0baS6M3ecNHbUFcPx5deXwcDiFKYPV6n5JciDm7e4yZ7z1yogcQ5nzLWZL7/pW3pyHkMNC4Nov8LjkjljfxIG888VXjcunE5yt7IQTdiQE0sWo3lrPj0vPWvqxVk3ZzasHE0WTB4un3kgqiqzfGL6oebPnyug9EZfLgeGHgZ8bIRwqPT9cvHoaKu4s6vvHpZ1aBzsHhztH4a5iBJG7Jf2dw8OpeORXt6XjaOj8tHuQS5irMYo7R8Ui0UZsVeMG/uHkuQjUmPsQwq74FguFuOsIAAyzUV4aow9fT6GflRKD5aPUGMclIvlcpTMgR43DmEiZT0XwdUY5eI8/VLSiH8JkI9Ixkh+KSC7LpZgSjKr4u7eQX6MKYwB1eMFBslbSmjZOczUk0HFuNcfoITEx2dDD3Pmcco8XjR7KbO3aOYpM180T1PmqTR/fLOh538hsdi4Ke3ouzul672N767/FP964qvCbwq/Lfy+oBcOC98V6oV2oV+ghVnhz4W/FP66/rf1v6//Y/2fseuXX8x/cfHrQmZb//d/AM0CVO0=</latexit>

R4camera

<latexit sha1_base64="y8EpttjuyGupqNEdLwvE88WHgMI=">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</latexit>

R4world

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R4robot

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R4head

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In the vertex shader:

Projection matrix

camera

coordinate

system

‣ field of view: angle �<latexit sha1_base64="sONfpeQh5hAcRlCpsOoQ/Yy2QjQ=">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</latexit>

‣ aspect ratio: a = widthheight

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‣ near clipping distance

‣ far clipping distancen> 0

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Projection matrix

camera

coordinate

system

(1,1,1)

(-1,-1,-1)

normalized device coordinate

2664

1a tan(�/2)

1tan(�/2)

� f +nf �n �

2 f nf �n

�1

3775

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How does it work?

Homogeneous coordinates

Brief History of Projective Geometry and homogeneous coordinates

Renaissance Art• Filippo Brunelleschi 1415: Linear perspective

Filippo Brunelleschi 1337–1446

‣ Found method accurate perspective drawing.

‣ Soon after Brunelleschi, most of the artists in Florence adopted the method.

‣ 2D representation of a 3D object?

Renaissance Art• The rules for representing 3D picture on a plane

‣ Parallel lines meet at the horizon.

‣ Image of a conic is a conic.

‣ Straight lines must be represented by straight lines.

Renaissance Art• We are exploring the collineations in the plane.

Renaissance Art• Live demo

Projective geometry• Girard Desargues 1600s‣ Projective Geometry: Geometry of straight edges.

Girard Desargues 1591–1661

‣ We must include “infinity.”

intersecting lines

where does the intersection go?

‣ Every pairs of line the this plane have an intersection!

Projective geometry‣ A projective plane is the standard plane union a projective line of infinity.

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‣ A projective line is a standard line union a point of infinity.

‣ Parallel lines in the plane meet at a point the line of infinity.

lines of the same slope,say slope = 0.5

0.5

‣ Parallel lines of “infinite slope” meet at the “additional point” of the projective line of infinity.

projective plane

projective line

Projective geometry

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lines of the same slope,say slope = 0.5

0.5

projective plane

projective line

intersecting lines

they still intersect

The 2D drawings are snapshots of a projective plane.

• Desargues’ projective geometrywasn’t appreciated in the 1600’s.

• It was not clear how to representinfinities arithmetically.

Taylor’s 1715 principles of perspective• Relation between the 3D object and 2D image

‣ Intersect the line on the image plane.‣ Join line from a fixed point (eye) to the 3D object.

‣ The image plane is thought of as Renaissance artists’ and Desargues’ projective plane where geometry of straight edges take place.

‣ Each point on the image plane is a line in 3D through the origin (eye).

‣ A line in 3D through the origin is a 3-proportion.

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plane

Between 2D position & 3D position

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Renaissance artists’ drawing canvas

points atinfinity

O 2 R3<latexit sha1_base64="WjRmOr+TDX1PFPjn6EvjRrwmRIk=">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</latexit>

Between 2D position & 3D position

lines in

canv

as2D subspaces in the

3D space

1D subspaces in the 3D space

points in canvas


Between 2D position & 3D positionEvery collineation in the drawing canvas

lines in

canv

as2D subspaces in the

3D space

1D subspaces in the 3D space

points in canvas


is just a 3D linear transformation.


ïp1

ò


ï�p�

ò


ïu0

ò


Renaissance artists’ drawing canvas

points atinfinity


These 3D representations for the 2D pointson the drawing canvas are the homogeneous coordinates.

Math historyThese 3D representations for the 2D pointson the drawing canvas are the homogeneous coordinates.

After the full introduction of the homogeneous coordinates in the 19th century (Möbius, Cayley), projective geometry becomes the most popular geometry.

In the 20th century, the most popular geometry has switched to differential geometry (Riemannian geometry). Projective geometry is no-longer a standard curriculum in mathematics.

Math historyIn the 20th century, the most popular geometry has switched to differential geometry (Riemannian geometry). Projective geometry is no-longer a standard curriculum in mathematics.

• Traces of the projective geometry can still be seen todayin linear algebra, algebraic geometry, and of course, computer graphics.

Back to our goal:Perform 3D collineation

We want a 3D collineation

2664

1a tan(�/2)

1tan(�/2)

� f +nf �n �

2 f nf �n

�1

3775


Renaissance artists:

Graphicsdevelopers:

linear transformin 3D

2D canvas 2D canvas

linear transformin 4D

3D world 3D NDC


In the homogeneous coordinate, we view each 4D vector as a 4-proportion.264

xyzw

375 ⇠

264�x�y�z�w

375

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We write to indicate that they have the same proportion.


264

26�45

375

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264

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⇠

264�100�300200�250

375

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2 : 6 : �4 : 5= 1 : 3 : �2 : 2.5= �100 : �300 : 200 : �250

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Between 3D position & 4D homo. coord.

24

pxpypz

35

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homogenize


24

pxpypz

35


264

pxpypz1

375




homogenize

264

pxpypzpw

375

<latexit sha1_base64="XE92FHlJu8CqNNNkgRysGQpr5bA=">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</latexit>

dehomogenize


24

pxpypz

35


264

pxpypz1

375




homogenize

264

pxpypzpw

375


dehomogenize

264

px/pw

py/pw

pz/pw

1

375

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⇠<latexit sha1_base64="zHKK1up7kk+0iyIePnWa2JTIesc=">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</latexit>

24

px/pw

py/pw

pz/pw

35

<latexit sha1_base64="102DR/H0DunY/pYdUt5pKPYMghk=">AAAilHiczZpbb9vKEceV01vq3pJToC994YlhtChsQ5RjWwaSg9iSbMkXSdbFdhIZwXK1kmjxdnYpxwpLoF+zr/0knSXFWZKSDBTowyFgav2bmd3Z5Z/Di2R4lin8YvHfL777xS9/9evfvPztxu9+/4c//unV6+9vhDvjlPWpa7n8ziCCWabD+r7pW+zO44zYhsVujWlF2m8fGRem6/T8ucfubTJ2zJFJiQ/oy6t/Dgw2Np3AsInPzadwY+CYlI04ocGgMtY178tTmLS+hoOBtmSfp+zaKodvKYeNAXOGONiXV5vF3WK0acsNfdHYLCy29pfXPxQGQ5fObOb41CJCfNaLnn8fEO6b1GLQ/Uwwj9ApGbPP0HSIzcR9EC1TqG0BGWojl8Of42sRTUcExBZibhvgCQlORN4m4Srb55k/Kt8HpuPNfObQeKDRzNJ8V5Nrrg1NzqhvzaFBKDchV41OCCyRD0cm05MYCUosNnzv8xnbhuZ7asN+W7qD830gBx8yYY6d7GTjsPtgwqxH5mdswZM8GtvJYcnaCKWwliIL5SC+61oSY1swP5C+jMuFyw4fqM630liYM8f0n7K+EkHPW5oAEUyZP+HubDzRbEInIGI+z8zLcbkNM7Pvgxnssx1R4lBmhRsbW1tat1477mgnrbtaFyTMvgryyAz3KRiMXBc+wwhS17YJ6G/wKCaMcMk/l+4/7+7fb2hasAU7TRvYgKMWogiLdG/BZilM2SZianqDr8OFWRm84UgmkgXMj+Uf6NqmroHO011ZFvECOcn0cGkH6IEz4bs86TUy5uY3+ZnMryhn+P+ZHxzjWrOaOc7A5KZVW/2Ty5p20jmuXNR6XanZKSNQCn0Q68agyqBUcHYK5/wpsU1rHrSu7sLgyunCue5atTAI0SlG0hXsl4SPF0SES0F2GMiToMv8Z4JgP1wONeLQVGLdCfHY6iGCaB3e7Rz8qGlo25fw3cHOYRoeRPBwp5yGhxEs7xylYTmCRzt68UcFjwC+04s7eikF9WJESzsQnqKljfSqrZ+BsXYGOyewNCunEVlWziWyrJxQZFk5q8iycmqRZfX8YtNikqAjKVtnbLHBh5kzZCMoU8OYc77MF4tyBYWzCtdl2wQVBkkPcC2UFdX1mJNSfWZbEtHf9IO34Wq6brRFXovRqOUK9j8Md6ivGg5ocmq5/gTOrMXpJ0/L1unSWbix9X55g6p/2mr24DRudXr142a1u9ptAKs5OD6O0zeM4DgMY3ZyguwkYZUKskrCqlVk1YTVashqCTs9RXaasLMzZGcJq9eR1RPWaCBrJOz8HNl5wi4ukF0k7PIS2WXCrq6QXSWs2UTWTFirhayVsHYbWTth19fIrhPW6SDrJKzbRdZNWK+HrJewfh9ZP2E3N8huEnZ7i+w2YXd3yO4S9vEjso8J+/QJ2SdgMTQSIYyUEIwTZCgEo4IMhWBUkaEQjBoyFIJxigyFYJwhQyEYdWQoBKOBDIVgnCNDIRgXyFAIxiUyFIJxhQyFYDSRoRCMFjIUgtFGhkIwrpGhEIwOMhSC0UWGQjB6yFAIRh8ZCsG4QYZCMG6RoRCMO2QoBOMjMhSC8QmZEgJdCAFudpUS6ImCKAVaURC1QKsKohhoTUFUAz1VEOVAzxREPdC6gigI2lAQFUHPFURJ0AsFURP0UkEUBb1SEFVBmwqiLGhLQdQFbSuIwqDXCqIyaEdBlAbtKojaoD0FURy0ryCqg94oiPKgtwqiPuidgigQ+lFBVAj9pKCSiFhIRFCuJCJOFESJiIqCKBFRVRAlImoKokTEqYIoEXGmIEpE1BVEiYiGgigRca4gSkRcKIgSEZcKokTElYIoEdFUECUiWgqiRERbQZSIuFYQJSI6CqJERFdBlIjoKYgSEX0FUSLiRkGUiLhVECUi7hREiYiPCqJExCcFU5cTgqWFYLkxkBnIKDKKbIhsiIwhY8hGI4QjhGNkY2QTZBNkJjIT2QOyB2RTZFNkFjILmY3MRuYgc5C5yFxkHjIP2U/IfkLGkXFkAplA5iPzkc2QzZA9IntE9hXZV2RPyJ6QzZHNkX1D9k0p4Rvjas5FyVfdjkb3rdf9416t02y0mqvvWeVda+rlhGHYYXyHLG+ZhUY04c8t8xsbavpWsq167jTiZ8Z+mLTszNOi8jy2vAkxWLKesXP8kX0b4DrJg0BkDXR44E62ODmVGkw0E2uaGGprZpg3PzykzA9L5uk0ZZ7m0oq7njLu6Lv79iyVoBmmeC4qHnFV1MMzUXEiq6KmmShtY2vF9owq2p3Wea3Sa9zUtG77uFJbr410Op02rkvQiUpt2lpJWStL1nrKWm8/K9mr415da7VrneNeq7M2M3DkjFjbmmmTsekQPt/WYCQtesjTPMJ9obkjzZnZRvS6krN0Og2Q5wAeZzk8FXL5vjVoRDUm69Vhea8Oy09MjJ28k0Rh5hm3tTAHA+IT8Jf7klwDTb4BJ1wj1pgZnMTnt8+1XI8+FqehPHMyNiCJsRadN2kjkMRYd+2cEcjaPK+IB0/TsIuDTTufkok1eSof3TM2IHgt84gjlnKWNPEw8zmbmDIZPuRsQPBGyh3ljEAiYWmu55u2+S36smDl/P4BB4KPbVMeivjzOTfyFLuRp+fcTGcUucnPZ9zEzIvc5KfM9dJk2pi7M09E8pX/LsQg1hyas8swgL811i5Yu2utbWluP2fvS3t/be8t6L211grBMnZrtVm4ixow4mQawH+hBpVXuDBhOJWHc20Yv4zaXdO9mGXiZ2vFO7bSjmNrraOwLJLpc71r1RzB8Y3262Yfu3Sf9fGk5qL9hjz6XUZdOOqjmTMk8ushYslveux16cr+G4MfGpFybhgFqlWIRWfWTGTL0piTIb61iz+5HUQ0HDhu9MpNLJWylUHi+Sg649ZyUETXxnDXXw6RMBORLcZD83E5RsK1o1xWcwGLx451/nUmxPIQEU3H5KY/YXSaW7iIwXraRExya//MOsqg7CTjfrKTzaYCl3LuGsQwLdOfx3Wx4j7m6iKQpGjekHytBhIXzdqTz7iZEZSM+MqGYwbllCV3xSbcgsv3xjwq65EFUo7cwmSY6L9oZqmgCVR9yoLi7j57CgecmPHXKTsxCLbDxGNHl/+rLiG5hrMyOT99G6nuzB0njR11xXB8+fVlMLA4henDVWpxCfLg5i1usqf8tQpInMMZc23my2/6Vp6cx1DDwiDar/G4ZM7Yn4TB4nON140Lp5PcrS1EEzbkxJLFaNFaTM9Lz5p6cdbNmQ0rR5MFk4fLZx6Iqsosn5h+qPmL5woovdGXy4GhhwEfGyEcKn2/XDw62i7u7ul7h2UdGgd7B0f7h2EuooQR+6X9vYND6Xikl/dl4+iofLR3kIsYqzFK+wfFYlFGvC3Gjf1DSfIRqTH2IYU9cCwXi3FWEACZ5iI8NcZbfTGGflRKD5aPUGMclIvlcpTMgR43DmEiZT0XwdUY5eIi/VLSiH8JkI9Ixkh+KSC7LpZgSjKr4t7bg/wYUxgDqsccBslbSmjZPczUk0HF+KzfQwmJj8+mHubM45R5vGz2UmZv2cxTZr5snqbMU2n+8mpTz/9CYrlxU9rV93ZL1283P1z/K/71xMvCXwtvCn8v6IXDwodCvdAu9Au08J8XL1+8fvH9m7+8efem8qYWu373YvGLiz8XMtub5n8B8uh6HQ==</latexit>

Between 3D position & 4D homo. coord.R3

<latexit sha1_base64="yriudDN+P+jnAyDbaHKk8PxYxII=">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</latexit>


264

pxpypzpw

375


dehomogenize

264

px/pw

py/pw

pz/pw

1

375

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⇠<latexit sha1_base64="zHKK1up7kk+0iyIePnWa2JTIesc=">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</latexit>

24

px/pw

py/pw

pz/pw

35

<latexit sha1_base64="102DR/H0DunY/pYdUt5pKPYMghk=">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</latexit>

264

pxpypz0

375

<latexit sha1_base64="Be1v9mtzo9/Rg6nmp5zlwxYIKC4=">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</latexit>

represents “points at infinity,” (or a pure direction like a vector)

(It is a good idea to keep positions represented in 4D, so that we can talkabout points at infinity at ease)


264

1234

375

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dehomogenize

24

0.250.5

0.75

35

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ïp1

ò


ï�p�

ò


ïu0

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physical space



lines in

physi

cal sp

ace

2D subspaces in the

homogeneous coord.

1D subspaces in the homogeneous coord.

points in physical space



lines in

physi

cal sp

ace

2D subspaces in the

homogeneous coord.

1D subspaces in the homogeneous coord.

points in physical space

Every collineation on the physical space24

pxpypz

35 7!

24

pnewx

pnewy

pnewz

35

<latexit sha1_base64="aGt+1+GWaIMHImTqGVgqqeG9pyQ=">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</latexit>

can be written as the result of

‣ Step1: homogenize

‣ Step2: 4D linear transformation

‣ Step3: dehomogenize

An example projective transformation264

11

11

375

<latexit sha1_base64="YpGwmltmSd2CLd+aa0qNOmrHtx8=">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</latexit>

will perform the following transformation:

Each point

24

pxpypz

35


is mapped to

24

pnewx

pnewy

pnewz

35 =?

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11

11

375

264

pxpypz1

375

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=

264

pxpy1pz

375

<latexit sha1_base64="dqwVOO1QiypyvVi6n+tPiw7tZtg=">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</latexit>

⇠

264

px/pz

py/pz

1/pz

1

375

<latexit sha1_base64="X6NCwCBoo+E0oAb7VE7c2+z3nNg=">AAAilHiczZpbb9vKEceV01vq3pJToC994YlhtChsQ5RjWwaSg9iSbMkXSdbFdhIZwXK1kmjxdnYpxwpLoF+zr/0knSXFWZKSDBTowyFgav2bmd3Z5Z/Di2R4lin8YvHfL777xS9/9evfvPztxu9+/4c//unV6+9vhDvjlPWpa7n8ziCCWabD+r7pW+zO44zYhsVujWlF2m8fGRem6/T8ucfubTJ2zJFJiQ/oy6t/DoRpbwwMNjadwLCJz82ncGPgmJSNOKHBoDLWNe/LU5i0voWDwZJ5njFryq5nDfrGgDlDHOfLq83ibjHatOWGvmhsFhZb+8vrHwqDoUtnNnN8ahEhPutFz78PCPdNajFIfCaYR+iUjNlnaDrEZuI+iJYp1LaADLWRy+HP8bWIpiMCYgsxtw3whAQnIm+TcJXt88wfle8D0/FmPnNoPNBoZmm+q8k114YmZ9S35tAglJuQq0YnBNbHhyOT6UmMBCUWG773+YxtQ/M9tWG/Ld3B+T6Qgw+ZMMdOdrJx2H0wYdYj8zO24Ekeiu3kmGRthFJYS5GFchDfdS2JsS2YH0hfxuXCZYcPVOdbaSzMmWP6T1lfiaDnLU2ACKbMn3B3Np5oNqETEDGfZ+bluNyGmdn3wQz22Y4ocSizwo2NrS2tW68dd7ST1l2tC+plXwV5ZIb7FAxGrgufYQSpa9sE9Dd4FBNGuOSfS/efd/fvNzQt2IKdpg1swFELUYRFurdgsxSmbBMxNb3B1+HCrAzecCQTyQLmx/IPdG1T10Dn6a4si3iBnGR6uLQD9MCZ8F2e9BoZc/Ob/EzmV5Qz/P/MD45xrVnNHGdgctOqrf7JZU076RxXLmq9rtTslBEohT6IdWNQZVAqODuFc/6U2KY1D1pXd2Fw5XThXHetWhiE6BQj6Qr2S8LHCyLCpSA7DORJ0GX+M0GwHy6HGnFoKrHuhHhs9RBBtA7vdg5+1DS07Uv47mDnMA0PIni4U07DwwiWd47SsBzBox29+KOCRwDf6cUdvZSCejGipR0IT9HSRnrV1s/AWDuDnRNYmpXTiCwr5xJZVk4osqycVWRZObXIsnp+sWkxSdCRlK0zttjgw8wZshGUqWHMOV/mi0W5gsJZheuybYIKg6QHuB7Kiup6zEmpPrMtiehv+sHbcDVdN9oir8Vo1HIF+x+GO9RXDQc0ObVcfwJn1uL0k6dl63TpLNzYer+8QdU/bTV7cBq3Or36cbPaXe02gNUcHB/H6RtGcByGMTs5QXaSsEoFWSVh1SqyasJqNWS1hJ2eIjtN2NkZsrOE1evI6glrNJA1EnZ+juw8YRcXyC4SdnmJ7DJhV1fIrhLWbCJrJqzVQtZKWLuNrJ2w62tk1wnrdJB1EtbtIusmrNdD1ktYv4+sn7CbG2Q3Cbu9RXabsLs7ZHcJ+/gR2ceEffqE7BOwGBqJEEZKCMYJMhSCUUGGQjCqyFAIRg0ZCsE4RYZCMM6QoRCMOjIUgtFAhkIwzpGhEIwLZCgE4xIZCsG4QoZCMJrIUAhGCxkKwWgjQyEY18hQCEYHGQrB6CJDIRg9ZCgEo48MhWDcIEMhGLfIUAjGHTIUgvERGQrB+IRMCYEuhAA3u0oJ9ERBlAKtKIhaoFUFUQy0piCqgZ4qiHKgZwqiHmhdQRQEbSiIiqDnCqIk6IWCqAl6qSCKgl4piKqgTQVRFrSlIOqCthVEYdBrBVEZtKMgSoN2FURt0J6CKA7aVxDVQW8URHnQWwVRH/ROQRQI/aggKoR+UlBJRCwkIihXEhEnCqJEREVBlIioKogSETUFUSLiVEGUiDhTECUi6gqiRERDQZSIOFcQJSIuFESJiEsFUSLiSkGUiGgqiBIRLQVRIqKtIEpEXCuIEhEdBVEioqsgSkT0FESJiL6CKBFxoyBKRNwqiBIRdwqiRMRHBVEi4pOCqcsJwdJCsNwYyAxkFBlFNkQ2RMaQMWSjEcIRwjGyMbIJsgkyE5mJ7AHZA7IpsikyC5mFzEZmI3OQOchcZC4yD5mH7CdkPyHjyDgygUwg85H5yGbIZsgekT0i+4rsK7InZE/I5sjmyL4h+6aU8I1xNeei5KtuR6P71uv+ca/WaTZazdX3rPKuNfVywjDsML5DlrfMQiOa8OeW+Y0NNX0r2VY9dxrxM2M/TFp25mlReR5b3oQYLFnP2Dn+yL4NcJ3kQSCyBjo8cCdbnJxKDSaaiTVNDLU1M8ybHx5S5ocl83SaMk9zacVdTxl39N19e5ZK0AxTPBcVj7gq6uGZqDiRVVHTTJS2sbVie0YV7U7rvFbpNW5qWrd9XKmt10Y6nU4b1yXoRKU2ba2krJUlaz1lrbeflezVca+utdq1znGv1VmbGThyRqxtzbTJ2HQIn29rMJIWPeRpHuG+0NyR5sxsI3pdyVk6nQbIcwCPsxyeCrl83xo0ohqT9eqwvFeH5Scmxk7eSaIw84zbWpiDAfEJ+Mt9Sa6BJt+AE64Ra8wMTuLz2+darkcfi9NQnjkZG5DEWIvOm7QRSGKsu3bOCGRtnlfEg6dp2MXBpp1PycSaPJWP7hkbELyWecQRSzlLmniY+ZxNTJkMH3I2IHgj5Y5yRiCRsDTX803b/BZ9WbByfv+AA8HHtikPRfz5nBt5it3I03NupjOK3OTnM25i5kVu8lPmemkybczdmSci+cp/F2IQaw7N2WUYwN8aaxes3bXWtjS3n7P3pb2/tvcW9N5aa4VgGbu12izcRQ0YcTIN4L9Qg8orXJgwnMrDuTaMX0btrulezDLxs7XiHVtpx7G11lFYFsn0ud61ao7g+Eb7dbOPXbrP+nhSc9F+Qx79LqMuHPXRzBkS+fUQseQ3Pfa6dGX/jcEPjUg5N4wC1SrEojNrJrJlaczJEN/axZ/cDiIaDhw3euUmlkrZyiDxfBSdcWs5KKJrY7jrL4dImInIFuOh+bgcI+HaUS6ruYDFY8c6/zoTYnmIiKZjctOfMDrNLVzEYD1tIia5tX9mHWVQdpJxP9nJZlOBSzl3DWKYlunP47pYcR9zdRFIUjRvSL5WA4mLZu3JZ9zMCEpGfGXDMYNyypK7YhNuweV7Yx6V9cgCKUduYTJM9F80s1TQBKo+ZUFxd589hQNOzPjrlJ0YBNth4rGjy/9Vl5Bcw1mZnJ++jVR35o6Txo66Yji+/PoyGFicwvThKrW4BHlw8xY32VP+WgUkzuGMuTbz5Td9K0/OY6hhYRDt13hcMmfsT8Jg8bnG68aF00nu1haiCRtyYslitGgtpuelZ029OOvmzIaVo8mCycPlMw9EVWWWT0w/1PzFcwWU3ujL5cDQw4CPjRAOlb5fLh4dbRd39/S9w7IOjYO9g6P9wzAXUcKI/dL+3sGhdDzSy/uycXRUPto7yEWM1Ril/YNisSgj3hbjxv6hJPmI1Bj7kMIeOJaLxTgrCIBMcxGeGuOtvhhDPyqlB8tHqDEOysVyOUrmQI8bhzCRsp6L4GqMcnGRfilpxL8EyEckYyS/FJBdF0swJZlVce/tQX6MKYwB1WMOg+QtJbTsHmbqyaBifNbvoYTEx2dTD3Pmcco8XjZ7KbO3bOYpM182T1PmqTR/ebWp538hsdy4Ke3qe7ul67ebH67/Ff964mXhr4U3hb8X9MJh4UOhXmgX+gVa+M+Lly9ev/j+zV/evHtTeVOLXb97sfjFxZ8Lme1N879Zn3ov</latexit>

dehomogenize

24

px/pz

py/pz

1/pz

35

<latexit sha1_base64="q10TkdpykdIIvIiVY7OYhGJ+qcA=">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</latexit>


11

11

375

<latexit sha1_base64="YpGwmltmSd2CLd+aa0qNOmrHtx8=">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</latexit>

will perform the following transformation:

Each point

24

pxpypz

35


is mapped to

24

pnewx

pnewy

pnewz

35 =

24

px/pz

py/pz

1/pz

35

<latexit sha1_base64="zC5CDnp0DGpob1ZO89K+6HfW+q4=">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</latexit>

Derivation of the projection matrix

Projection matrix

camera

coordinate

system

(1,1,1)

(-1,-1,-1)

normalized device coordinate

2664

1a tan(�/2)

1tan(�/2)

� f +nf �n �

2 f nf �n

�1

3775


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cse 167 (fa21) computer graphics: projective geometry

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