Triangles to Pixels - Computerphile

Key Takeaways

we have taken our 3D object specified its vertices relative to an origin in a coordinate reference frame what happens to a triangle to turn it from this representation that we've got the vertices the faces into pixels on the screen we've got faces we can now transform them um by applying um a transformation at every vertex so we can turn them and scale them and move them around um but how does that now fit into the bigger picture how do we go from the local space of our 3D object in which its vertices are defined into a 2d representation on a screen well there are several stages and they're all trans different Transformations that we apply the first one is um from the local object space of um our representation into World space so we pick an origin for the world which I suppose we'll say is the middle of the table and we then apply a transformation which respecifies all of the vertices of our 3D object in local space in terms of the world origin so what we're basically doing is saying well we want our pyramid to be at um uh plus one in in X or minus one in X um minus one in zed 2 in y uh we can also then rotate it and uh scale it if we like and we can do that with several objects and move them all to different places and each will have a different transformation which places it um in the world it moves it from its local space to World space and that is called a world space transformation what we can also do because we can combine Transformations together is specify one object in terms of another so that we can Nest these different Transformations so that we might want to stick this pyramid to that one um and so it has first has a transformation that moves it from its own local space to the space of this pyramid which then moves to the the world space wherever we want to place it in the world and so we can nest things that way all we need to know really is where each vertex is in the world we need to know it World space position all of the vertices then need to be respecified from World space in terms of the space of the Viewpoint if the Viewpoint is here say this is our little camera and it's looking out on the world as we turn the camera this way what we actually do is apply a transformation to everything in the world which swings it over this way so the camera really never moves all we're doing is applying a transformation that appears to move everything relative to the camera we're respecifying all of our objects in terms of the local space of the camera and this is view space or eye space if you like so it's really it's just another transformation all we're doing is moving from World space into view space where're respecifying one coordinate frame of reference in terms of another so just like we moved from object space the local space of uh the pyramid to World space where it was positioned in our world um we're now moving from World space into view space which is again just another transformation and we can actually just combine those Transformations and have a a world view transformation which combines both of them into one and so all we need to do is apply that operation at every vertex on every single shape we then need to project down from 3D space into 2D space which again is just another transformation everything is just another transformation a projective transformation is a little bit different to the others because whereas before we were just applying translations and scales and rotations to move objects from one space to the next to the next now what we have to do is have a special transformation um which removes all of the depth information from the scene and squashes all of these vertices down onto a 2d plane and also because uh we might want to have a perspective effect whereby objects in the distance seem smaller we need to collapse and scale them down based on their distance and so on and so forth and all of that is encoded in the projective transformation so yeah so what have we done we've we've taken our local representation moved it into World space constructed a scene out of our pyramids and uh then moved all of those relative to the eye so we know exactly what's in front of us and what's going to be seen and then we collapse all of those down um into two Dimensions so what happens to a triangle to turn it from this representation that we've got the vertices the faces into pixels on the screen well let's just draw some pixels out not very many pixels the resolution is a bit low so let's just take a random triangle and draw it on let's draw it in purple so here's our representation now it's been squashed down into two dimensions and what we now have to do to turn it into pixels is go through every pixel here and decide whether or not it's inside the triangle now I've not drawn very many pixels on here so it's not going to look very good but we can say that um this one is pretty much inside the triangle um this one is pretty much inside the triangle and for argument sake I'll go with this one and this one and these two as well and so now what we've done is taken that simple geometric representation and discretized it we've turned it into a representation that no longer has any information about um vertices and faces it just says this pixel is inside this triangle when we collapse down from 3D to 2D um some of the vertices are going to lie outside our field of view and in that case we need to clip them what we have to actually do is generate another two vertices here because we don't know what's happening to this because it's not part of our representation anymore because we have clipped it to uh the viewport as it's called the problem is now as you can see we've no longer got a triangle so well we just use our old trick of turning an odd polygon into a triangle representation into two triangles and then the process of deciding which pixels are in the Triangle proceeds but for these two new triangles that now fit perfectly inside our Viewpoint and that's called clipping and then The Next Step Beyond that is just coloring in the pixels and deciding exactly uh how to shade them and light them we'd like to thank audible.com for their support of this computer file video and if you'd like to download one of their huge range of books go to audible.com computer file and you can download one for free I'd like to recommend one today um my favorite uh computer related book and one of the first books that made me think hang on computers could be cool and that's Neuromancer by William Gibson so get on to audible.com computer file download your free book and thanks again to audible.com for supporting this computer file video we'll incorporate the table into this because this is where the shadow is being cast the table is another 3D object in our 3D World and we're doing a a sheer operation in four dimensional space