From 95c33c221cb151b794fe32f0761014287ec5561f Mon Sep 17 00:00:00 2001 From: Mouse <937279156@qq.com> Date: Mon, 24 Sep 2018 21:03:37 +0800 Subject: [PATCH] Update 15-terrain-collisions --- source/.vscode/settings.json | 3 + source/15-terrain-collisions.md | 271 ++++++++++++++++++ source/_static/15/cell.png | Bin 0 -> 8963 bytes .../_static/15/model_to_world_coordinates.png | Bin 0 -> 35072 bytes .../_static/15/terrain_block_variables_n.png | Bin 0 -> 32897 bytes source/_static/15/terrain_grid.png | Bin 0 -> 13535 bytes source/_static/15/tunnelling.png | Bin 0 -> 8377 bytes source/index.rst | 1 + 8 files changed, 275 insertions(+) create mode 100644 source/.vscode/settings.json create mode 100644 source/15-terrain-collisions.md create mode 100644 source/_static/15/cell.png create mode 100644 source/_static/15/model_to_world_coordinates.png create mode 100644 source/_static/15/terrain_block_variables_n.png create mode 100644 source/_static/15/terrain_grid.png create mode 100644 source/_static/15/tunnelling.png diff --git a/source/.vscode/settings.json b/source/.vscode/settings.json new file mode 100644 index 0000000..6e57044 --- /dev/null +++ b/source/.vscode/settings.json @@ -0,0 +1,3 @@ +{ + "restructuredtext.confPath": "d:\\Workspace\\lwjglbook-CN-Translation\\source" +} \ No newline at end of file diff --git a/source/15-terrain-collisions.md b/source/15-terrain-collisions.md new file mode 100644 index 0000000..50165ee --- /dev/null +++ b/source/15-terrain-collisions.md @@ -0,0 +1,271 @@ +# 地形碰撞 (Terrain Collisions) + +此前我们创建了一个地形,接下来就是检测碰撞以避免穿过它。回忆一下之前的内容,一个地形是由地形块组成的,每个地形块都是由高度图生成的,高度图用于设置构成地形的三角形的顶点高度。 + +为了检测碰撞,我们必须将当前所在位置的**Y**值与当前地形点的**Y**值进行比较。如果有碰撞,我们需要回到地形上方。很简单的想法,是吗?确实是这样,但在比较之前,我们需要进行几次计算。 + +我们首先要定义的是我们对“当前位置”这个词的理解。由于我们还没有一个球员的概念,答案很简单,当前的位置将是相机的位置。因此,我们已经有了比较的组成部分之一,因此,接下来要计算的是当前位置的地形高度。 +首先要定义的是“当前位置”这个词的概念。由于我们还没有一个“玩家”的概念,因此当前位置将是摄像机的位置。这样我们就有了比较的一方,因此接下来要计算当前位置的地形高度。 + +如上所是,地形由地形块组成,如下图所示。 + +![地形网格](_static/15/terrain_grid.png) + +每个地形块都是由相同的高度图网格构成,但被精确地缩放和位移,以形成看起来像是连续的景观的地形网格。 + +所以首先要做的是确定当前位置(摄像机位置)在哪个地形块。为了得到它,我们将基于位移和缩放来计算每个地形块的包围盒(**BoundingBox**)。因为地形在运行时不会移动或缩放,所以我们可以在`Terrain`类的构造方法中计算。这样就可以在任何时候访问它们,而不需要在每个游戏循环周期中重复这些计算。 + +我们将创建一个新的方法来计算一个地形块的包围盒,名为`getBoundingBox`。 + +```java +private Box2D getBoundingBox(GameItem terrainBlock) { + float scale = terrainBlock.getScale(); + Vector3f position = terrainBlock.getPosition(); + + float topLeftX = HeightMapMesh.STARTX * scale + position.x; + float topLeftZ = HeightMapMesh.STARTZ * scale + position.z; + float width = Math.abs(HeightMapMesh.STARTX * 2) * scale; + float height = Math.abs(HeightMapMesh.STARTZ * 2) * scale; + Box2D boundingBox = new Box2D(topLeftX, topLeftZ, width, height); + return boundingBox; +} +``` + +`Box2D`是`java.awt.Rectangle2D.Float`类的简化版本,为了避免使用AWT而创建。 + +限制我们需要计算地形块的世界坐标。在上一章中,你看到所有的地形网格都是在一个正方形中创建的,它的原点设置为`[STARTX, STARTZ]`。因此,我们需要把这些坐标转换为世界坐标,这要考虑下图所示的位移与缩放。 + +![模型坐标系到世界坐标系](_static/15/model_to_world_coordinates.png) + +如上所述,这可以在`Terrain`类构造方法中计算,因为它不会在运行时发生变化,所以我们要添加一个新的属性来保存包围盒: + +```java +private final Box2D[][] boundingBoxes; +``` + +在`Terrain`类的构造方法中,当我们创建地形块时,只需调用计算包围盒的方法。 + +```java +public Terrain(int terrainSize, float scale, float minY, float maxY, String heightMapFile, String textureFile, int textInc) throws Exception { + this.terrainSize = terrainSize; + gameItems = new GameItem[terrainSize * terrainSize]; + + PNGDecoder decoder = new PNGDecoder(getClass().getResourceAsStream(heightMapFile)); + int height = decoder.getHeight(); + int width = decoder.getWidth(); + ByteBuffer buf = ByteBuffer.allocateDirect( + 4 * decoder.getWidth() * decoder.getHeight()); + decoder.decode(buf, decoder.getWidth() * 4, PNGDecoder.Format.RGBA); + buf.flip(); + + // 每行与每列的顶点数 + verticesPerCol = heightMapImage.getWidth(); + verticesPerRow = heightMapImage.getHeight(); + + heightMapMesh = new HeightMapMesh(minY, maxY, buf, width, textureFile, textInc); + boundingBoxes = new Box2D[terrainSize][terrainSize]; + for (int row = 0; row < terrainSize; row++) { + for (int col = 0; col < terrainSize; col++) { + float xDisplacement = (col - ((float) terrainSize - 1) / (float) 2) * scale * HeightMapMesh.getXLength(); + float zDisplacement = (row - ((float) terrainSize - 1) / (float) 2) * scale * HeightMapMesh.getZLength(); + + GameItem terrainBlock = new GameItem(heightMapMesh.getMesh()); + terrainBlock.setScale(scale); + terrainBlock.setPosition(xDisplacement, 0, zDisplacement); + gameItems[row * terrainSize + col] = terrainBlock; + + boundingBoxes[row][col] = getBoundingBox(terrainBlock); + } + } +} +``` + +因此,有了所有预先计算的包围盒,我们将创建一个新的方法,这个方法将以当前位置为参数,返回对应地形高度。该方法名为`getHeight`,其定义如下。 + +```java +public float getHeight(Vector3f position) { + float result = Float.MIN_VALUE; + // 对于每个地形块,我们获取包围盒,将其转换到观察坐标系 + // 检查坐标是否包含在包围盒中 + Box2D boundingBox = null; + boolean found = false; + GameItem terrainBlock = null; + for (int row = 0; row < terrainSize && !found; row++) { + for (int col = 0; col < terrainSize && !found; col++) { + terrainBlock = gameItems[row * terrainSize + col]; + boundingBox = boundingBoxes[row][col]; + found = boundingBox.contains(position.x, position.z); + } + } + + // 如果我们找到了一个包含我们位置的地形块 + // 计算该位置的地形高度 + if (found) { + Vector3f[] triangle = getTriangle(position, boundingBox, terrainBlock); + result = interpolateHeight(triangle[0], triangle[1], triangle[2], position.x, position.z); + } + + return result; +} +``` + +在此方法中第一件事是确定我们所在的地形块。由于我们已经有了每个地形块的包围盒,所以算法很简单。我们只需要迭代包围盒数组,并检查当前位置是否位于其中(`Box2D`提供了该方法)。 + +一旦找到了地形块,我们需要计算所处的三角形,这是由之后的`getTriangle`方法计算的。之后,我们得到了所在三角形的坐标,包括它的高度。但是,我们需要的是一个点的高度,这个点不位于这些顶点中的任何一点,而位于它们之间的位置。这将在`interpolateHeight`方法中计算,我们也将解释这是如何计算的。 + +让我们先从确定所处的三角形开始。构成地形块的正方形可以看作一个网格,其中每个单元由两个三角形组成。首先我们定义一些变量: + +* **boundingBox.x**是与包围盒相关联的地形块的原**x**坐标。 +* **boundingBox.y**是与包围盒相关联的地形块的原**z**坐标(即使你看到一个**y**,但它是在**z**轴的)。 +* **boundingBox.width**是地形块正方形的宽度。 +* **boundingBox.height**是地形块正方形的高度。 +* **cellWidth**是一个单元的宽度。 +* **cellHeight**是一个单元的高度。 + +上面定义的所有变量都用世界坐标来表示。为了计算单元的宽度,我们只需要将包围盒宽度除以每列的顶点数: + +$$cellWidth = \frac{boundingBox.width}{verticesPerCol}$$ + +`cellHeight`的计算也相似: + +$$cellHeight = \frac{boundingBox.height}{verticesPerRow}$$ + +一旦有了这些变量,我们就可以计算所在的单元格的行和列了: + +$$col = \frac{position.x - boundingBox.x}{boundingBox.width}$$ + +$$row = \frac{position.z - boundingBox.y}{boundingBox.height}$$ + +下图在示例地形块展示了此前描述的所有变量。 + +![地形块变量](_static/15/terrain_block_variables_n.png) + +有了这些信息,就可以计算单元格中包含的三角形顶点的位置。我们怎么才能做到呢?让我们来看看组成一个单元格的三角形。 + +![单元格](_static/15/cell.png) + +你可以看到,单元格是被一个对角线分开为两个三角形的。确定与当前位置相关的三角形的方法,是检查**z**坐标在对角线的上方还是下方。在本例中,将对角线的**x**值设置为当前位置的**x**值,求出对应的对角线**z**值,如果当前位置的**z**值小于对角线的**z**值,那么我们在**T1**中。反之如果当前位置的**z**值大于对角线的**z**值,我们就在**T2**中。 + +我们可以通过计算与对角线相匹配的直线方程来确定。 + +如果你还记得学校的数学课,从两点通过的直线(在二维中)的方程为: + +$$y-y1=m\cdot(x-x1)$$ + +其中m是直线的斜率,也就是说,当沿**x**轴移动时,其高度会发生变化。请注意,在本例中,**y**坐标其实是一个**z**。还要注意的是,我们使用的是二维坐标,因为在这里不计算高度,只要**x**坐标和**z**坐标就足够了。因此,在本例中,直线方程应该是这样。 + +$$z-z1=m\cdot(x-x1)$$ + +斜率可以按如下方式计算: + +$$m=\frac{z1-z2}{x1-x2}$$ + +所以给定一个**x**坐标得到一个**z**值的对角线方程就像这样: + +$$z=m\cdot(xpos-x1)+z1=\frac{z1-z2}{x1-x2}\cdot(zpos-x1)+z1$$ + +其中**x1**、**x2**、**z1**和**z2**分别是顶点**V1**和**V2**的**x**和**z**坐标。 + +因此,通过上述方式来获得当前位置所在的三角形的方法,名为`getTriangle`,其实现如下: + +```java +protected Vector3f[] getTriangle(Vector3f position, Box2D boundingBox, GameItem terrainBlock) { + // 获得与当前位置相关的高度图的行列 + float cellWidth = boundingBox.width / (float) verticesPerCol; + float cellHeight = boundingBox.height / (float) verticesPerRow; + int col = (int) ((position.x - boundingBox.x) / cellWidth); + int row = (int) ((position.z - boundingBox.y) / cellHeight); + + Vector3f[] triangle = new Vector3f[3]; + triangle[1] = new Vector3f( + boundingBox.x + col * cellWidth, + getWorldHeight(row + 1, col, terrainBlock), + boundingBox.y + (row + 1) * cellHeight); + triangle[2] = new Vector3f( + boundingBox.x + (col + 1) * cellWidth, + getWorldHeight(row, col + 1, terrainBlock), + boundingBox.y + row * cellHeight); + if (position.z < getDiagonalZCoord(triangle[1].x, triangle[1].z, triangle[2].x, triangle[2].z, position.x)) { + triangle[0] = new Vector3f( + boundingBox.x + col * cellWidth, + getWorldHeight(row, col, terrainBlock), + boundingBox.y + row * cellHeight); + } else { + triangle[0] = new Vector3f( + boundingBox.x + (col + 1) * cellWidth, + getWorldHeight(row + 2, col + 1, terrainBlock), + boundingBox.y + (row + 1) * cellHeight); + } + + return triangle; +} + +protected float getDiagonalZCoord(float x1, float z1, float x2, float z2, float x) { + float z = ((z1 - z2) / (x1 - x2)) * (x - x1) + z1; + return z; +} + +protected float getWorldHeight(int row, int col, GameItem gameItem) { + float y = heightMapMesh.getHeight(row, col); + return y * gameItem.getScale() + gameItem.getPosition().y; +} +``` + +你可以看到我们有另外两个反复。第一个名为`getDiagonalZCoord`,给定**x**位置和两个顶点计算对角线的**z**坐标。另一个名为`getWorldHeight`,用来获得三角形顶点的高度(即**y**坐标)。当地形网格被创建时,每个顶点的高度都被预先计算和储存,我们只需将其转换为世界坐标。 + +好,我们有当前位置的三角形坐标。最后,我们准备在当前位置计算地形高度。怎么做呢?我们的三角形在一个平面上,一个平面可以由三个点定义,在本例中,三个顶点定义了一个三角形。 + +平面方程如下: + +$$a\cdot x+b\cdot y+c\cdot z+d=0$$ + +上述方程的常数值是: + +$$a=(B_{y}-A_{y}) \cdot (C_{z} - A_{z}) - (C_{y} - A_{y}) \cdot (B_{z}-A_{z})$$ + +$$b=(B_{z}-A_{z}) \cdot (C_{x} - A_{x}) - (C_{z} - A_{z}) \cdot (B_{z}-A_{z})$$ + +$$c=(B_{x}-A_{x}) \cdot (C_{y} - A_{y}) - (C_{x} - A_{x}) \cdot (B_{y}-A_{y})$$ + +其中**A**、**B**和**C**是定义平面所需的三个顶点。 + +然后,利用之前的方程以及当前位置的**x**和**z**坐标值,我们能够计算**y**值,即当前位置的地形高度: + +$$y = (-d - a \cdot x - c \cdot z) / b$$ + +实现了如上运算的方法如下: + +```java +protected float interpolateHeight(Vector3f pA, Vector3f pB, Vector3f pC, float x, float z) { + // 平面方程 ax+by+cz+d=0 + float a = (pB.y - pA.y) * (pC.z - pA.z) - (pC.y - pA.y) * (pB.z - pA.z); + float b = (pB.z - pA.z) * (pC.x - pA.x) - (pC.z - pA.z) * (pB.x - pA.x); + float c = (pB.x - pA.x) * (pC.y - pA.y) - (pC.x - pA.x) * (pB.y - pA.y); + float d = -(a * pA.x + b * pA.y + c * pA.z); + // y = (-d -ax -cz) / b + float y = (-d - a * x - c * z) / b; + return y; +} +``` + +这就完了!现在我们能够检测碰撞,所以在`DummyGame`类中,在更新摄像机位置时,修改如下代码: + +```java +// 更新摄像机位置 +Vector3f prevPos = new Vector3f(camera.getPosition()); +camera.movePosition(cameraInc.x * CAMERA_POS_STEP, cameraInc.y * CAMERA_POS_STEP, cameraInc.z * CAMERA_POS_STEP); +// 检查是否发生碰撞。如果为true,将y坐标设置为 +// 最大高度 +float height = terrain.getHeight(camera.getPosition()); +if ( camera.getPosition().y <= height ) { + camera.setPosition(prevPos.x, prevPos.y, prevPos.z); +} +``` + +如你所见,检测地形碰撞的概念很容易理解,但是我们需要仔细地进行计算并了解正处理的不同坐标系。 + +此外,虽然这里给出的算法在大多数情况下都是可用的,但仍存在需要仔细处理的情况。你可以发现的一个问题是隧道效应(`Tunnelling`)。设想一个情况,我们正以高速穿过地形,正因如此,位置增量值较高。这个值变得如此之高,以至于因为我们检测的是最终位置的碰撞,所以可能已经穿过了位于两点之间的障碍。 + +![隧道效应](_static/15/tunnelling.png) + +有许多可行的解决方案可以避免这个效应,最简单的解决方法是将要进行的计算分成增量较小的多份。 diff --git a/source/_static/15/cell.png b/source/_static/15/cell.png new file mode 100644 index 0000000000000000000000000000000000000000..e04738919cce7c815594019cb6fba1c555e9f2fc GIT binary patch literal 8963 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