全球圖形學(xué)領(lǐng)域教育的領(lǐng)先者、自研引擎的倡導(dǎo)者、底層技術(shù)研究領(lǐng)域的技術(shù)公開者，東漢書院在致力于使得更多人群具備內(nèi)核級(jí)競爭力的道路上，將帶給小伙伴們更多的公開技術(shù)教學(xué)和視頻，感謝一路以來有你的支持。我們正在用實(shí)際行動(dòng)來幫助小伙伴們構(gòu)建一套成體系的圖形學(xué)知識(shí)架構(gòu)，你在我們這里獲得的不止于那些毫無意義的代碼，我們這里更多的是代碼背后的故事，以及精準(zhǔn)、透徹的理解。我們不會(huì)扔給人們一本書或者給個(gè)思路讓人們?nèi)プ詫W(xué)，我們是親自來設(shè)計(jì)出好的課程，讓人們明白到底背后還有哪些細(xì)節(jié)。

這里插播一個(gè)引擎大賽的消息，感興趣的同學(xué)可以看一眼，這也是東漢書院的立項(xiàng)使命：

東漢書院：自研引擎大賽?zhuanlan.zhihu.com

In the displacement mapping example, all we did was use a (very large) texture to drive displacement from a flat surface and then use tessellation to increase the number of polygons in the scene. This is a type of brute-force, data-driven approach to geometric complexity. In the cubicbezier example described here, we will use math to drive geometry—we’re going to render a cubic Bézier patch. If you revisit Chapter 4, “Math for 3D Graphics”, you’ll see that we’ve covered all the number crunching we’ll need here.

在置換貼圖的例子中，我們所做的事情就是用一個(gè)紋理去使得一個(gè)平面進(jìn)行偏移并且使用tessellation技術(shù)來增加場景中的幾何形體的個(gè)數(shù)。這是一種比較簡單粗暴的玩法。在cubicbezier的例子中，我們 將會(huì)使用數(shù)學(xué)知識(shí)來構(gòu)建幾何形體。如果你回顧一下第四章的內(nèi)容，你將會(huì)看到我們?nèi)绾蝸硎褂媚切├碚摗?/p>

A cubic Bézier patch is a type of higher-order surface and is defined by control points that provide input to the interpolation functions that define the surface’s shape. A Bézier patch has 16 control points, laid out in a 4 × 4 grid. Very often (including in this example), those control points are equally spaced in two dimensions, varying only in distance from a shared plane. However, they don’t have to be. Free-form Bézier patches are an extremely powerful modeling tool being used natively by many pieces of modeling and design software. With OpenGL tessellation, it’s possible to render them directly.

一個(gè)cubic bezier patch是一種高階曲面，它是由一些控制點(diǎn)來定義的，我們可以通過這些控制點(diǎn)插值出來幾何體表面上的各個(gè)點(diǎn)。一個(gè)bezier patch有16個(gè)控制點(diǎn)，組成了一個(gè)4x4的網(wǎng)格。通常情況下， 那些控制點(diǎn)之間是等間距，變化的部分僅僅是到共享平面的距離。然而這并不重要。在建模軟件中，自由形式的bezier patch是非常好的建模工具。通過OpenGL的tessellation技術(shù)，我們可以直接渲染出他們來。

The simplest method of rendering a Bézier patch is to treat the four control points in each row of the patch as the control points for a single cubic Bézier curve, just as was described in Chapter 4. Given our 4 × 4 grid of control points, we have four curves; if we interpolate along each of them using the same value of t, we will end up with four new points. We use these four points as the control points for a second cubic Bézier curve. Interpolating along this second curve using a new value for t gives us a second point that lies on the patch. The two values of t (let’s call them t0 and t1) are the domain of the patch and are handed to us by the tessellation evaluation shader in http://gl_TessCoord.xy.In this example, we’ll perform tessellation in view space. That is, in our vertex shader, we’ll transform our patch’s control points into view space by multiplying their coordinates by the model–view matrix. This simple vertex shader is shown in Listing 8.12.

就如同第四章介紹的那樣，最簡單的渲染bezier patch的方法就是把每一行的四個(gè)控制點(diǎn)都當(dāng)成是一個(gè)cubic bezier曲線的控制點(diǎn)。如此一來，給我們4x4個(gè)控制點(diǎn)之后，我們就可以得到四條曲線。如果我們 對(duì)它們進(jìn)行插值的都是使用同一個(gè)插值因子t，那么我們將會(huì)得到四個(gè)新的點(diǎn)。我們使用這四個(gè)點(diǎn)產(chǎn)生第二個(gè)cubic bezier曲線。在我們第二條cubic bezier曲線進(jìn)行插值的時(shí)候，我們使用另外的一個(gè)t值進(jìn)行插值，這樣 我們就可以拿到第二批點(diǎn)。這倆t值是存在于patch領(lǐng)域下的，并且通過tessellation evaluation shader的gl_TessCoord.xy傳給我們。在本例子中，我們?cè)谝暱诳臻g進(jìn)行曲面細(xì)分。也就是說，我們的vertex shader中， 我們要把我們的patch的控制點(diǎn)都轉(zhuǎn)到視口空間里來。清單8.12展示了我們的vertex shader。

#version 450 core in vec4 position; uniform mat4 mv_matrix; void main(void){gl_Position = mv_matrix * position; }

Listing 8.12: Cubic Bézier patch vertex shader

Once our control points are in view space, they are passed to our tessellation control shader. In a more advanced algorithm, we could project the control points into screen space, determine the length of the curve, and set the tessellation factors appropriately. However, in this example, we’ll settle for a simple fixed tessellation factor. As in previous examples, we set the tessellation factor only when gl_InvocationID is zero, but pass all of the other data through once per invocation. The tessellation control shader is shown in Listing 8.13.

當(dāng)我們的控制點(diǎn)都轉(zhuǎn)到視口空間下來了之后，他們將會(huì)被發(fā)送給tessellation control shader。在一些更進(jìn)一步的算法中，我們可以把這些控制點(diǎn)投影到屏幕空間里去，這樣就可以求得 曲線的長度，然后就可以設(shè)置合適的細(xì)分因子了。然而在本例子中，我們只是簡單的設(shè)置一個(gè)固定的細(xì)分因子。如同前面的那些例子，我們僅當(dāng)gl_InstanceID等于0的時(shí)候才去設(shè)置細(xì)分因子， 把其他數(shù)據(jù)簡單粗暴的傳給下一個(gè)shader階段。清單8.13展示了tessellation control shader。

#version 450 corelayout (vertices = 16) out; void main(void){if (gl_InvocationID == 0){gl_TessLevelInner[0] = 16.0;gl_TessLevelInner[1] = 16.0;gl_TessLevelOuter[0] = 16.0;gl_TessLevelOuter[1] = 16.0;gl_TessLevelOuter[2] = 16.0;gl_TessLevelOuter[3] = 16.0;}gl_out[gl_InvocationID].gl_Position = gl_in[gl_InvocationID].gl_Position; }

Listing 8.13: Cubic Bézier patch tessellation control shader

Next, we come to the tessellation evaluation shader, which is where the meat of the algorithm is found. The shader in its entirety is shown in Listing 8.14. You should recognize the cubic_bezier and quadratic_bezier functions from Chapter 4. The evaluate_patch function is responsible for evaluating the vertex’s coordinate given the input patch coordinates and the vertex’s position within the patch.

下一步，我們來到tessellation evaluation shader里面了，在這里我們實(shí)現(xiàn)核心的算法部分。清單8.14展示了全部的代碼清單。你應(yīng)該可以看得明白cubic_bezier和quadratic_bezier 函數(shù)了，因?yàn)榈?章介紹過的。evaluate_patch函數(shù)就是用來根據(jù)輸入的patch的坐標(biāo)和在patch里的那些點(diǎn)的位置來生成頂點(diǎn)的坐標(biāo)的。

#version 450 core layout (quads, equal_spacing, cw) in; uniform mat4 mv_matrix; uniform mat4 proj_matrix; out TES_OUT{vec3 N; } tes_out; vec4 quadratic_bezier(vec4 A, vec4 B, vec4 C, float t){vec4 D = mix(A, B, t);vec4 E = mix(B, C, t);return mix(D, E, t); } vec4 cubic_bezier(vec4 A, vec4 B, vec4 C, vec4 D, float t){vec4 E = mix(A, B, t);vec4 F = mix(B, C, t);vec4 G = mix(C, D, t);return quadratic_bezier(E, F, G, t); } vec4 evaluate_patch(vec2 at){vec4 P[4];int i;for (i = 0; i < 4; i++){P[i] = cubic_bezier(gl_in[i + 0].gl_Position,gl_in[i + 4].gl_Position,gl_in[i + 8].gl_Position,gl_in[i + 12].gl_Position,at.y);}return cubic_bezier(P[0], P[1], P[2], P[3], at.x); } const float epsilon = 0.001; void main(void){vec4 p1 = evaluate_patch(gl_TessCoord.xy);vec4 p2 = evaluate_patch(gl_TessCoord.xy + vec2(0.0, epsilon));vec4 p3 = evaluate_patch(gl_TessCoord.xy + vec2(epsilon, 0.0));vec3 v1 = normalize(p2.xyz - p1.xyz);vec3 v2 = normalize(p3.xyz - p1.xyz);tes_out.N = cross(v1, v2);gl_Position = proj_matrix * p1; }

Listing 8.14: Cubic Bézier patch tessellation evaluation shader

In our tessellation evaluation shader, we calculate the surface normal to the patch by evaluating the patch position at two points very close to the point under consideration, using the additional points to calculate two vectors that lie on the patch and then taking their cross product. This result is passed to the fragment shader shown in Listing 8.15.

在我們的tessellation evaluation shader中，我們先根據(jù)patch中的兩個(gè)點(diǎn)計(jì)算出這種條件下額外的一個(gè)點(diǎn)，然后用這額外的一個(gè)點(diǎn)構(gòu)成2個(gè)在這個(gè)patch上的向量，然后叉乘出來法線。這個(gè)結(jié)果會(huì)傳給 清單8.15展示的fragment shader。

#version 450 coreout vec4 color; in TES_OUT{vec3 N; }fs_in; void main(void){vec3 N = normalize(fs_in.N);vec4 c = vec4(1.0, -1.0, 0.0, 0.0) * N.z + vec4(0.0, 0.0, 0.0, 1.0);color = clamp(c, vec4(0.0), vec4(1.0)); }

Listing 8.15: Cubic Bézier patch fragment shader

This fragment shader performs a very simple lighting calculation using the z component of the surface normal. The result of rendering with this shader is shown in Figure 8.14.

這個(gè)fragment shader根據(jù)表面的法線的z分量簡單粗暴的計(jì)算出了光照。圖8.14展示了渲染出來的結(jié)果圖像。

Because the rendered patch shown in Figure 8.14 is smooth, it is difficult to see the tessellation that has been applied to the shape. The left side of Figure 8.15 shows a wireframe representation of the tessellated patch, and the right side of Figure 8.15 shows the patch’s control points and the control cage, which is formed by creating a grid of lines between the control points.

因?yàn)閳D8.14展示出來的渲染出來的patch比較光滑，所以很難看出細(xì)分效果。在圖8.15中，我們?cè)谧筮呎故玖司€框模式下的結(jié)果畫面，右邊的圖中我們展示了那些生成畫面用的那些控制點(diǎn)，以及控制點(diǎn)之間的連線。

我們核心關(guān)注和討論的領(lǐng)域是引擎的底層技術(shù)以及商業(yè)化方面的信息，可能并不適合初級(jí)入門的同學(xué)。官方相關(guān)信息主要包括了對(duì)市面上的引擎的各種視角的分析以及宏觀方面的技術(shù)討論，相關(guān)領(lǐng)域感興趣的同學(xué)可以關(guān)注東漢書院以及圖形之心公眾號(hào)。

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