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----------------------------------------------------------------------------- VGOPHONG.TXT - Phong lighting and specular highlights. Theory, practice and explaination of the phong lighting and shading model. ----------------------------------------------------------------------------- by TimJ/Vertigo "I am he, as you are he, as you are me, and we are all together" email: tim@legend.co.uk irc: #coders #vertcode revision history: 16/02/97 v1.0 - Initial version ----------------------------------------------------------------------------- INTRODUCTION ----------------------------------------------------------------------------- First off, I hope this doc is of use to some people, and maybe other people will find it interesting. Recently, I've been thinking a lot about phong shading and lighting. There was something that was bugging me. I couldn't quite put my finger on it. It was something I knew to be true, but I had to explain it to myself. It all started when I was chatting to Vector about true phong shading. We'd both recently looked at Voltaire/OTM's doc on fast phong shading (again). We were (among other things) chatting about lighting functions in our 3D engine. It was about specular highlights and the way normal (fast) phong lighting doesn't yield specular highlights on a flat plane (all normals pointing one way). This got me thinking. I thought, well of course it doesn't, because the light calculated at each point will be the same (because all normals are the same). This annoyed me because I knew it wasn't true -- but I couldn't remember why. It's all to do with the light vector and the view vector (and keeping them constant). Volatire's phong method also doesn't yield specular highlights in the center of polygons. Oh, and his method emulates the equation given exactly -- it's not a tradeoff. But then again, it's also just the same as using gouraud. I'll explain later. Then I remembered the actual theory behind the lighting equations. It stuck me that people tend to get mixed up in code and forget about the theory behind it. What I'll do is go through the theory, how it's implemented. Based on that I'll then address what this doc is actually about -- highlights in the center of polygons and on flat planes. If you think this doc is a bit slow, forgive me, but you can never please everybody :) [Oh, important point.. this is generally about the phong equation. You can do it per pixel or per vertex, either way it's the phong equation. Just because you gouraud shade doesn't mean you can't have phong style specular highlights.] ----------------------------------------------------------------------------- LIGHT RAY REFLECTION ----------------------------------------------------------------------------- I was going to explain the theory behind light ray relfection and the spread of the ray across a surface depending on the angle of incidence. But but then I realized I'd have to go into they physics behind spectral reflectivity too, so I won't :) If there's enough demand for it, email me and I'll put it in. ----------------------------------------------------------------------------- PHONG'S SPECULAR HIGHLIGHTS ----------------------------------------------------------------------------- We need to understand how the phong lighting equation is made up. Let's define a few useful values: ^N | L | R V \ | / __/ \ | / __/ \ | / __/ \ | /__/ \|// -------------.-------------- P ^ point under consideration It's important you know what these values actually are: N = surface normal L = unit vector between point and light V = unit vector between point and view R = light reflection unit vector (mirror of L about N) First, the diffuse relfection is given by the Lamertian Relfection equation: diffuse = Kd * (N dot L) Where Kd is the diffuse relfection constant. (N dot L) is the same as the cosine of the angle between N and L, so as the angle decrease, the resulting diffuse value is higher. Phong gave spectral reflectivity as: diffuse + Ks * (R dot V)^n Which is: Kd * (N dot L) + Ks * (R dot V)^n Where Kd is the diffuse component and Ks is the specular compoenet. This is the generally accepted phong lighting equation. Ks is generaly taken to be a specularity constant (although Phong defined it as W(i).. see later). As the angle between the view (V) and the reflected light (R) decreases, you will get more specularity. The clever thing about Phong's equation was that it gave a neat way to calculate the specular intensity 'bump' around the light reflection vector (R). The larger the exponential power (n) the smaller and more intense the specular intensity bump. Hence specular highlights. ----------------------------------------------------------------------------- IMPLEMENTATION OF PHONG'S EQUATION ----------------------------------------------------------------------------- Most people simplify this equation somewhat, for speed. We begin with : Kd * (N dot L) + Ks * (R dot V)^n The obvious thing we'd like to remove is (R dot V). Since we don't want to calculate the light relfection vector (mirror of light incidence around the surface normal) -- because it's expensive. Blinn introduced a way to do this using an imaginary vector H. It's then reduced to (N dot H). H is defined as halfway between L and V (after L and V are normalized). H is therefore (L + V) / 2. You will see that the angle R dot V is double N dot H -- but this doesn't matter as you can alter the specular exponential value (n) to compenstate. This gives us the equation : Kd * (N dot L) + Ks * (N dot ( L + V / 2))^n Up until now we've ignored the ambient factor, this is because it's damn obvious and has little consequence on the math.. we'll put it in now Ka + Kd * (N dot L) + Ks * (N dot ( L + V / 2))^n Which is easily implemented. You only need three vectors: the surface normal, the light vector and the view vector. It's obviously advised to do this equation in object-space. Another way to remove R dot V, is by replacing it with N dot L : Ka + Kd * (N dot L) + Ks * (N dot L)^n This assumes you will always get the maximum specularly reflected light, no matter where the view is. Here's why : If we assume V is always the same as R, then the angle between N and V is the same as N and L -- ^N | A = angle between N and L L | R (also V) B = angle between N and V \ A | B / \ /|\ / \/ | \/ \ | / \|/ -------------.-------------- Angle A and B are the same (of course, since R is the mirror vector of L). So, N dot V becomes the same as L dot N. This makes life easier and faster. The results completely ignore the position of the view; so it's like having a reflective surface that always reflects the maximum amount of specular light towards the view. (normally, as the angle between the view and the reflected light increases, you get less specularly reflected light). It's just a trade off. Ka + Kd * (N dot L) + Ks * (N dot L)^n or Ka + Kd * cos(theta) + Ks * cos(theta)^n where cos(theta) is N dot L. Most likely, the above equation is the one most people use. Also, since more implementation assume V is constant across the scene (normalized.. at infinity) then using N dot L can be acceptable. But it does have some dire consequences. ----------------------------------------------------------------------------- REAL TIME PHONG SHADING ----------------------------------------------------------------------------- This is what was causing the confusion. Voltaires text on phong shading (OTMPHONG.TXT) used the equation color = specular + (cos x) * diffuse + (cos x)^n * specular for calculating phong lighting. This is the same as the last equation we just disussed. He then went on to explain the specular intensity 'bump' through the (cos x)^n term of the equation. (note: phong shading is done by recalculating the lighting equation at each pixel -- this is done by interpolating the vertex normals across the polygon and re-evaluating). Since there is just one angle term in the equation (N dot L), he realized he could dispense with normals and just interpolate the angle. Remember : Ka + Kd * cos(theta) + Ks * cos(theta)^n You only need to interpolate theta, then you can do a lookup table for the correct colour created like so: for( theta=0 ; theta < 90 ; theta++ ) { table[i] = Ka + Kd * cos(theta) + Ks * pow( cos(theta) , specExp ) ; } The problem with all this is that the original equation was inaccurate, so the results will be inaccurate. However, Voltaire does point this out, and state that highlights can't be inside polygons. But, as Zog pointed out to me, you can get exactly the same effect with gouraud, by setting up the palette in a similar way. This method is basically the same as gouraud, you're just interpolating an angle instead of an intensity.. as Zog put it, "it's fucking gouraud revisited" :) I just thought I'd clear this up, as people tend not to believe it's real, and think it's some kind of trick (if you use the same equation in a true shader, you'll get the same results). It's also important for the next section (the equation at least). ----------------------------------------------------------------------------- PHONG'S SPECULAR HIGHLIGHTS (REVISITED) ----------------------------------------------------------------------------- As pointed out, phong shading requires the interpolating of vertex normals across the polygon, and recalculation of the equation. Right, here's where more confustion comes in. To simplify things, people tend to treat V as a constant over the entire scene. N L | / V (view) (light) \ | / \ | / \ | / \|/ -------------.-------------- P V is not constant though.. it is dependant on the point under consideration (P). So making V constant, is like sticking the view at infinity (this is done by normalizing the view vector). This means that the falloff of the specular light at sharp angles between the surface and view is not taken into account (it's linear). So the highlight will be too big and intense at sharp angles (the falloff will be linear in respect to the view position). Also, the highlight won't move correctly with the view. You can probably see that the same thing can be done for L.. ie. directional lighting. Putting L at infinity affects specular fall off with respect to the light and the view. But I'm sure everyone knows the implications of directional lighting (it's just like having a light really really far away). As explained earlier, dispensing with V altogether can give you a nice speed up using N dot L instead of R dot V. Let's look at the consequences of dispensing with V. This is like assuming you have a perfect reflector.. like a mirror. The surface will always reflect the maximum amount of specular light towards the view -- the highlight will seem to 'stick' to the light relfection vector and not change shape or size, no matter where you put the view. But, given this, you just have to interpolate N for the polygon (remember, we need (N dot L) and (N dot H) where is H = L+V/2) ----------------------------------------------------------------------------- SPECULAR REFLECTION ----------------------------------------------------------------------------- Now we get to the crux of the problem. All that time ago back at the start, I mentioned a plane with all the normals pointing one way. The thing to remember is this, the lighting is *not* just dependant on the surface normal. It's a function of the light vector and the view vector. The important value is V, as V affects how specular light is reflected. If V is properly calculated, for a flat surface, the angle between the view and the normal (which is constant) will alter. N V ^ \ \ | \ \ | \ \ | \ \ | \ \ | \ \ | -------.-----------.--- p1 p2 The angle at p1 is obviously sharper than the angle at p2. Even though N is the same at both points. It all becomes obvious now. Unless you calculate V correctly, the reflected specular light over a flat surface will be the even at any point on it. At this point you might be thinking : "what if V is put at infinity and L is calculated properly? -- won't that do the same job?" In a word, no, because V is the important vector. We have to remember the original equation where the specular light was a function of R and V. N would be constant, V would be constant, so specular light would just be a function of V -- ie, not very accurate at all. So, we're left with the following equation: Ka + Kd * (N dot L) + Ks * (N dot ( L + V / 2))^n So, basically, in a nutshell, you've got to recalculate V for the new point under consideration. If you do that, you'll get specular highlights on flat planes. If you have directional lights (L at infinity) then the highlight will be incorrectly positioned and spread -- since the angle between any given point and the light will be more or less the same (to an infintecial degree). On a flat surface you'd be depending on the angle the view makes with the surface. Although, this isn't too much to worry about -- I think directional lighting is a choice, not a compromise. [ Oh yeah, in the first section I said Phong defined Ks as W(i). Well, this meant that Ks was a function of the angle of incidence between the light and the surface. So specularly reflected light was dependant on the incoming angle as well as the outgoing angle. Phong never actually defined W(i) though -- so it's usually ignored. It does give you another parameter for your surface to play with though. ] ----------------------------------------------------------------------------- REAL TIME PHONG SHADING (REVISIED) ----------------------------------------------------------------------------- Ok, so now we know that V and L are important factors of the equation. Voltaire's phong shading method is completely correct for the equation he used (but as mentioned, it's basically the same as gouraud). What he did was place L at infinity and make the surface a perfect reflector (that always reflected the maximum specular light) towards the view (where ever it was). I explained all this earlier anyway :) That meant there was only one angle to interpolate, but it also meant that on flat surfaces it was impossible to get correct highlights. (polygons are flat :) However, Voltaire's method is more accurate than normal gouraud for the same lighting equation (due to the non-linear lookup table). However, you can do the same with gouraud.. it just another way of implenenting it (and quite redundant). His method won't extend to other equations though. What's annoying is when people just say his method doesn't work and is a load of crap, without explaining why. ----------------------------------------------------------------------------- CLOSING WORDS "picture yourself in boat on river with tangerine trees and marmalade skies" ----------------------------------------------------------------------------- Well, I guess I've managed to confuse almost eveyone. What I've tried to do is explain the factors of the phong lighting equation -- what parts do what, and why it works. I've probably made loads of typo's and got different bit's mixed up -- but hell, I don't care :) With any luck, I might not get flamed. If I have mixed something up, forgive me -- it get's hard to track what you have/have not said etc.. and it's not easy without some good diagrams :) There are a couple of sections I've left out.. I started the one on light ray reflection, but left it out. I also has a section on optimizing the equation : Ka + Kd * (N dot L) + Ks * (N dot ( L + V / 2))^n But I didn't get it finished (it's based on the routines I use in my 3D engine). There are no square roots, tables, divides or pow()'s used. Yet it still produces the same results (to a reasonable degree of accuracy -- any errors are covered up by (a). the number of intensities we can actually display and (b). the precision of the fpu). If there's enough demand then I'll put it in. At the end of all this, an interesting thing to note is that all these equations have no physical basic what so ever -- they're just equations that fit real work observations. Light is actually better represented as radiation -- but let's not get into that now :) Greets: Oh god, I don't know.. um.. people I know.. hmmm (no particular order) Vector Vastator Phred Gooroo Midnight aM Eckart codex PGM (where ever he may be) BigCheese Pel Wog (Zog, whatever) Crom God All at Abstract Entertainment [plus anyone I've missed] Flames/comments go to: tim@legend.co.uk ----------------------------------------------------------------------------- "Living is easy with eyes closed. Misunderstanding all you see. It's getting hard to be someone, but it all works out. It doesn't matter much to me."