计算机图形学-变换基础

坐标系转换历程
模型坐标系 -> 世界坐标系 -> 摄像机坐标系 -> 视口（屏幕）坐标系

变换

仿射变换和线性变换
线性：旋转 缩放 镜像 切变
放射： 平移

平移

2D变换矩阵

3D变换矩阵

旋转

2D旋转矩阵

 //2D 旋转
        private (float,float) VertexRotate(float angle ,float x,float y)
        {
            float newX = (float)(x * Math.Cos(angle) - y * Math.Sin(angle));
            float newY = (float)(x * Math.Sin(angle) + y * Math.Cos(angle));
            return (newX,newY);
        }

        public void Rotate(int degree)
        {
            float angle = (float)(degree / 360.0f * Math.PI);
            //a点
            float newX, newY;
            (newX,newY) =  VertexRotate(angle, A.X, A.Y);
            A.X = newX;
            A.Y = newY;
            
            //b点
            (newX, newY) = VertexRotate(angle, B.X, B.Y);
            B.X = newX;
            B.Y = newY;
            
            //C点
            (newX, newY) = VertexRotate(angle, C.X, C.Y);
            C.X = newX;
            C.Y = newY;
        }

e.Graphics.TranslateTransform(200, 200);调整坐标中心到屏幕中央

3D 旋转矩阵

组合变换（平移 + 旋转）

dx dy dz是平移分量
矩阵乘积表示 组合变换比如
先旋转 再平移

可以看出左上角是旋转变换，最后一行是平移的变换

 //矩阵相乘
        public Matrix4X4 Mul(Matrix4X4 m)
        {
            Matrix4X4 newM = new Matrix4X4();
            for (int w = 1; w <= 4; w++)
                for (int h = 1; h <= 4; h++)
                    for (int n = 1; n <= 4; n++)
                    {
                        newM[w, h] += this[w, n] * m[n, h];
                    }
             return newM;
        }

        public Vector4 Mul(Vector4 v)
        {
            Vector4 newV = new Vector4();
            newV.x = v.x * this[1,1] + v.y * this[2,1] + v.z * this[3,1] + v.w * this[4,1];
            newV.x = v.x * this[1,2] + v.y * this[2,2] + v.z * this[3,2] + v.w * this[4,2];
            newV.x = v.x * this[1,3] + v.y * this[2,3] + v.z * this[3,3] + v.w * this[4,3];
            newV.x = v.x * this[1,4] + v.y * this[2,4] + v.z * this[3,4] + v.w * this[4,4];

            return newV;
        }

 //三角形利用矩阵乘法进行变换
        public void Transform(Matrix4X4 m)
        {
            this.a = this.A = m.Mul(this.A);
            this.b = this.B = m.Mul(this.B);
            this.c = this.C = m.Mul(this.C);
        }

        public void Draw(Graphics g)
        {
            
            g.DrawLines(new Pen(Color.Red,2),this.Get2DPointFArr());
        }

        private PointF[] Get2DPointFArr()
        {
            PointF[] arr = new PointF[4];
            arr[0] = Get2DPointF(this.a);
            arr[1] = Get2DPointF(this.b);
            arr[2] = Get2DPointF(this.c);
            arr[3] = arr[0];
            return arr;
        }

        private PointF Get2DPointF(Vector4 v)
        {
            PointF p = new PointF();
            p.X = (float)(v.x / v.w);
            p.Y = (float)(v.y / v.w);
            return p;
        }

此时只有单纯的旋转。
如果在Z方向上添加透视投影矩阵

   m_view = new Matrix4X4();
            m_view = [1, 1] = 1;
            m_view = [2, 2] = 1;
            m_view = [3, 3] = 1;
            m_view = [4, 3] = 250;
            m_view = [4, 4] = 1;

 m_projection = new Matrix4X4();
            m_projection = [1, 1] = 1;
            m_projection = [2, 2] = 1;
            m_projection = [3, 3] = 1;
            m_projection = [3, 4] = 1.0 / 250;

即可。

透视投影

小孔成像。利用相似三角形计算物体轮廓

透视投影矩阵

最终只有x y表达，实际的x 坐标需要 透视除法 （x / (z/d)）

撤销变换

变换矩阵A * 变换矩阵A的逆矩阵