参考:
https://juejin.im/post/58ce7fe561ff4b006c9a63c6
https://juejin.im/post/58c3b88a44d904006984e58d
https://github.com/lguipeng/BubbleView
https://ghui.me/post/2015/10/android-graphics-path/
http://www.jianshu.com/p/d6b4a9ad022e
弹性圆的绘制:http://www.jianshu.com/p/791d3a791ec2
google api demo
PathMeasure
http://blog.csdn.net/u013831257/article/details/51565591
贝塞尔曲线:
https://juejin.im/post/58ce7f0461ff4b006c9a5e66
Path##
Path 表示路径,可使用Canvas.drawPath方法将其绘制出来,Path不仅可以使用Paint的填充模式和描边模式,也可以用画布裁剪和或者画文字。
Path 是Android中一种绘制工具,表示路径,可以画各种复杂的图形,如:曲线、轮廓等等;
基本方法##
**1. moveTo(x, y) ** : 移动到起点;
** 2. lineTo(x, y) **: 绘制直线;
3. arcTo(rectF, startAngle,sweepAngle): **
在RectF中绘制弧度(rect为弧度的外切rect), 从起点角度start,绘制sweep个 角度;
** 4. arcTo(rectF, startAngle, sweepAngle, true):表示在rect绘制弧度,并且与 moveTo 的起点分离,类似于另开一个Path;文档解释为:
If true, always begin a new contour with the arc
mPath.reset();
RectF rect = new RectF(100, 100, 200, 200);
mPaint.setColor(Color.RED);
canvas.drawRect(rect, mPaint);
mPaint.setColor(Color.GREEN);
mPath.arcTo(rect, 0, 90);
canvas.drawPath(mPath, mPaint);
效果图:
**5. reset(): ** 表示清空 path;
**6. addOval(rectF, direction): ** 添加一个椭圆,rectF为此椭圆的外切rect;
canvas.translate(200, 200);
RectF oval = new RectF(100, 100, 300, 400);
mPaint.setColor(Color.RED);
canvas.drawRect(oval, mPaint);
mPaint.setColor(Color.GREEN);
mPath.addOval(oval, Path.Direction.CW);// 外,CCW,内
canvas.drawPath(mPath, mPaint);
// 绘制路径上的文字
canvas.drawTextOnPath("sdjfsklfjksiewjf快睡觉富士康福克斯司法局斯柯达", mPath, 0, 0, mTextPaint);
canvas.restore();
效果图:
**7.addCircle(x, y, radius,Direction) : ** 绘制圆形;
**8.addPath(path) : ** 添加新的path到当前path;
9.quadTo(x1,y1,x2,y2) : 绘制二次贝塞尔曲线,其中 (x1,y1)为控制点,(x2,y2)为终点
// 曲线
mPath.moveTo(50, 50);
// 贝塞尔曲线(二阶)
mPath.quadTo(200, 200, 300, 100); // 以(200,200)为控制点,画贝塞尔曲线
canvas.drawPath(mPath, mPaint);
图示:
10.cubicTo(x1, y1, x2, y2, x3, y3):三阶贝塞尔,其中(x1,y1),(x2,y2)为控制点,(x3,y3)为终点
rXXX方法##
rXXX与原方法的区别是:r方法是基于当前绘制开始点的offest,比如当前paint位于 (100,100)处,则使用rLineTo(100,100)方法绘制出来的直线是从(100,100)到(200,200)的一条直线,由此可见rXXX方法方便用来基于之前的绘制作连续绘制。
mPath.reset();
canvas.translate(500, 500);
mPath.moveTo(100, 100);
mPath.lineTo(400, 400);
mPath.lineTo(400, 100);
mPath.rLineTo(100, 300); // mPath.LineTo(100, 300)
canvas.drawPath(mPath, mPaint);
示例:
Path.op方法(API >= 19 )##
用于将2个对象path做相应的运算组合,有点类似于数学上的集合运算
canvas.translate(0, 200);
Path path1 = new Path();
path1.addCircle(150, 150, 100, Path.Direction.CW);
Path path2 = new Path();
path2.addCircle(200, 200, 100, Path.Direction.CW);
path1.op(path2, Path.Op.INTERSECT); // (相交的部分)Union (inclusive-or) the two paths
canvas.drawPath(path1, mPaint);
图示:
总结如下:
- Path.Op.DIFFERENCE 减去path1中path1与path2都存在的部分;
path1 = (path1 - path1 ∩ path2) - Path.Op.INTERSECT 保留path1与path2共同的部分;
path1 = path1 ∩ path2 - Path.Op.UNION 取path1与path2的并集;
path1 = path1 ∪ path2 - Path.Op.REVERSE_DIFFERENCE 与DIFFERENCE刚好相反;
path1 = path2 - (path1 ∩ path2) - Path.Op.XOR 与INTERSECT刚好相反;
path1 = (path1 ∪ path2) - (path1 ∩ path2)
path的例子:圆角矩形##
在Android 中可用 canvas.drawRoundRect 来实现圆角矩形,如果不怕麻烦,
也可以使用 path 来实现;
如下例子:
private static final float DEFAULT_RADIUS = 50;
@Override
protected void onDraw(Canvas canvas) {
super.onDraw(canvas);
drawRect(canvas);
}
private void drawRect(Canvas canvas) {
RectF rectF = null;
// 圆弧起点
mPath.moveTo(100 + DEFAULT_RADIUS, 100);
mPath.lineTo(500 - DEFAULT_RADIUS, 100);
// 右上角圆角 (从 rectF 的中间开始画, path 为路径所以会连起来)
rectF = new RectF(500 - DEFAULT_RADIUS, 100, 500, (100 + DEFAULT_RADIUS));
mPath.arcTo(rectF, 270, 90);
// 右下圆角
mPath.lineTo(500, 300 - DEFAULT_RADIUS);
rectF = new RectF(500 - DEFAULT_RADIUS, 300 - DEFAULT_RADIUS, 500, 300);
mPath.arcTo(rectF, 0, 90);
// 左下圆角
mPath.lineTo(100 + DEFAULT_RADIUS, 300);
rectF = new RectF(100, 300 - DEFAULT_RADIUS, 100 + DEFAULT_RADIUS, 300);
mPath.arcTo(rectF, 90, 90);
// 左上圆角
mPath.lineTo(100, 100 + DEFAULT_RADIUS);
rectF = new RectF(100, 100, 100 + DEFAULT_RADIUS, 100 + DEFAULT_RADIUS);
mPath.arcTo(rectF, 180, 90);
// mPath.lineTo(100, 300);
mPath.close();
canvas.drawPath(mPath, mPaint);
}
效果图:
聊天气泡形状##
这用drawRoundRect就不能实现了。使用path,简单修改一下就可以了
//。。。。省略
// 左下圆角
mPath.lineTo(100 + DEFAULT_RADIUS, 300);
rectF = new RectF(100, 300 - DEFAULT_RADIUS, 100 + DEFAULT_RADIUS, 300);
mPath.arcTo(rectF, 90, 90);
// 小三角形,
mPath.lineTo(100, 100 + DEFAULT_RADIUS + 50);
mPath.lineTo(100 - 50, 100 + DEFAULT_RADIUS + 25);
mPath.lineTo(100, 100 + DEFAULT_RADIUS);
// 左上圆角
rectF = new RectF(100, 100, 100 + DEFAULT_RADIUS, 100 + DEFAULT_RADIUS);
mPath.arcTo(rectF, 180, 90);
mPath.close();
canvas.drawPath(mPath, mPaint);
效果图:
贝塞尔曲线##
贝塞尔曲线需要三个点才能确定,所以quadTo方法中的四个参数分别是确定第二,第三的点的。第一个点就是path上次操作的点。
画一条波浪线:
int WAVE_HEIGHT = 88;
public WaveView(Context context, @Nullable AttributeSet attrs, int defStyleAttr) {
super(context, attrs, defStyleAttr);
mPaint = new Paint(Paint.ANTI_ALIAS_FLAG);
mPaint.setColor(Color.RED);
mPaint.setStyle(Paint.Style.STROKE);
mPath = new Path();
}
@Override
protected void onDraw(Canvas canvas) {
super.onDraw(canvas);
mWL = getWidth();
mCenterY = getHeight() / 2;
startWave();
// 屏幕外 画第一断波纹的第一条曲线
mPath.moveTo(-mWL, mCenterY);
mPath.quadTo(-mWL * 3 / 4, mCenterY + WAVE_HEIGHT, -mWL / 2, mCenterY);
// 屏幕外 第一段波纹的第二天曲线
mPath.quadTo(-mWL / 4, mCenterY - WAVE_HEIGHT, 0, mCenterY);
// 屏幕内 画第二断波纹的第一条曲线
mPath.quadTo(mWL / 4, mCenterY + WAVE_HEIGHT, mWL / 2, mCenterY);
// 屏幕内 画第二断波纹的第二条曲线
mPath.quadTo(mWL * 3 / 4, mCenterY - WAVE_HEIGHT, mWL, mCenterY);
canvas.drawPath(mPath, mPaint);
}
效果图:
让曲线向右平移,就可以实现波浪效果了,让曲线上每个点的x加入offset平移值(不断绘制来实现动画);
private void startWave() {
ValueAnimator animator = ValueAnimator.ofInt(0, mWL); //mWL是一段波纹的长度
animator.setDuration(1000);
animator.setRepeatCount(ValueAnimator.INFINITE);
animator.setInterpolator(new LinearInterpolator());
animator.addUpdateListener(new ValueAnimator.AnimatorUpdateListener() {
@Override
public void onAnimationUpdate(ValueAnimator animation) {
offset = (Integer) animation.getAnimatedValue(); //offset 的值的范围在[0,mWL]之间。
postInvalidate();
}
});
animator.start();
}
// onDraw方法中
...
mPath.reset();
// 使用 for 循环来画
mPath.moveTo(-mWL + offset, mCenterY); // 移动起始点
for (int i = 0; i < 2; i++) {
mPath.quadTo(-mWL * 3 / 4 + (i * mWL) + offset, mCenterY + WAVE_HEIGHT, -mWL / 2 + (i * mWL) + offset, mCenterY);
mPath.quadTo(-mWL / 4 + (i * mWL) + offset, mCenterY - WAVE_HEIGHT, 0 + (i * mWL) + offset, mCenterY);
}
canvas.drawPath(mPath, mPaint);
贴一下完整代码:
public class WaveView extends View {
private Path mPath;
private Paint mPaint;
private int mWL;
private int mCenterY;
private final int WAVE_HEIGHT = 88;
public WaveView(Context context) {
this(context, null);
}
public WaveView(Context context, @Nullable AttributeSet attrs) {
this(context, attrs, 0);
}
public WaveView(Context context, @Nullable AttributeSet attrs, int defStyleAttr) {
super(context, attrs, defStyleAttr);
mPaint = new Paint(Paint.ANTI_ALIAS_FLAG);
mPaint.setColor(Color.RED);
mPaint.setStyle(Paint.Style.STROKE);
mPath = new Path();
}
int offset = 0;
private void startWave() {
ValueAnimator animator = ValueAnimator.ofInt(0, mWL); //mWL是一段波纹的长度
animator.setDuration(1000);
animator.setRepeatCount(ValueAnimator.INFINITE);
animator.setInterpolator(new LinearInterpolator());
animator.addUpdateListener(new ValueAnimator.AnimatorUpdateListener() {
@Override
public void onAnimationUpdate(ValueAnimator animation) {
offset = (Integer) animation.getAnimatedValue(); //offset 的值的范围在[0,mWL]之间。
postInvalidate();
}
});
animator.start();
}
@Override
protected void onDraw(Canvas canvas) {
super.onDraw(canvas);
if (mWL == 0) {
mWL = getWidth();
mCenterY = getHeight() / 2;
startWave();
}
// // 屏幕外 画第一断波纹的第一条曲线
// mPath.moveTo(-mWL, mCenterY);
// mPath.quadTo(-mWL * 3 / 4, mCenterY + WAVE_HEIGHT, -mWL / 2, mCenterY);
// // 屏幕外 第一段波纹的第二天曲线
// mPath.quadTo(-mWL / 4, mCenterY - WAVE_HEIGHT, 0, mCenterY);
//
// // 屏幕内 画第二断波纹的第一条曲线
// mPath.quadTo(mWL / 4, mCenterY + WAVE_HEIGHT, mWL / 2, mCenterY);
// // 屏幕内 画第二断波纹的第二条曲线
// mPath.quadTo(mWL * 3 / 4, mCenterY - WAVE_HEIGHT, mWL, mCenterY);
mPath.reset();
// 使用 for 循环来画
mPath.moveTo(-mWL + offset, mCenterY); // 移动起始点
for (int i = 0; i < 2; i++) {
mPath.quadTo(-mWL * 3 / 4 + (i * mWL) + offset, mCenterY + WAVE_HEIGHT, -mWL / 2 + (i * mWL) + offset, mCenterY);
mPath.quadTo(-mWL / 4 + (i * mWL) + offset, mCenterY - WAVE_HEIGHT, 0 + (i * mWL) + offset, mCenterY);
}
canvas.drawPath(mPath, mPaint);
}
}
cubicTo 三级曲线的使用###
使用4条三阶曲线来画一个正圆:
private void drawCircle() {
mPath.reset();
// 4条三阶曲线来表示圆
// 右上1/4圆
mPath.lineTo(0, -radius);
mPath.cubicTo(radius * sMagicNumber, -radius
, radius, -radius * sMagicNumber
, radius, 0);
mPath.lineTo(0, 0);
// 右下1/4圆
mPath.lineTo(0, radius);
mPath.cubicTo(radius * sMagicNumber, radius
, radius, radius * sMagicNumber
, radius, 0);
mPath.lineTo(0, 0);
// 左上1/4圆
mPath.lineTo(0, -radius);
mPath.cubicTo(-radius * sMagicNumber, -radius
, -radius, -radius * sMagicNumber
, -radius, 0);
mPath.lineTo(0, 0);
// 左下1/4圆
mPath.lineTo(0, radius);
mPath.cubicTo(-radius * sMagicNumber, radius
, -radius, radius * sMagicNumber
, -radius, 0);
mPath.lineTo(0, 0);
}
@Override
protected void onDraw(Canvas canvas) {
super.onDraw(canvas);
canvas.save();
canvas.translate(getWidth() / 2, getHeight() / 2);
canvas.drawPath(mPath, mPaint);
canvas.restore();
}
效果图:
确定点来生成圆####
如上图,共12个点,如果要画曲线P0P3,需要知道 我们就要知道 P0,P1,P2,P3 这四个点的坐标,P0,P3 已经知道了,分别为P0(0, -r), P3(r, 0);
p1 与 p2,如上,已经算出来了为:P1(rmagic, -r), P2(r, -rmagic);
点的封装
// 魔力数字
private float c = 0.551915024494f;
class HorizontalLine {
public PointF left = new PointF(); //P7 P11
public PointF middle = new PointF(); //P0 P6
public PointF right = new PointF(); //P1 P5
public HorizontalLine(float x, float y) {
left.x = -radius * c;
left.y = y;
middle.x = x;
middle.y = y;
right.x = radius * c;
right.y = y;
}
public void setY(float y) {
left.y = y;
middle.y = y;
right.y = y;
}
}
class VerticalLine {
public PointF top = new PointF(); //P2 P10
public PointF middle = new PointF(); //P3 P9
public PointF bottom = new PointF(); //P4 P8
public VerticalLine(float x, float y) {
top.x = x;
top.y = -radius * c;
middle.x = x;
middle.y = y;
bottom.x = x;
bottom.y = radius * c;
}
public void setX(float x) {
top.x = x;
middle.x = x;
bottom.x = x;
}
}
画圆,并与onTouchEvent绑定:
// 实现圆
private void updatePath(float x, float y) {
float distance = distance(mPrevX, mPrevY, x, y);
float longRadius = radius + distance;
float shortRadius = radius - distance * 0.1f;
mDegree = points2Degrees(mPrevX, mPrevY, x, y);
// 修改相关参数,缩放
mRightLine.setX(longRadius);
mLeftLine.setX(-shortRadius);
mTopLine.setY(-shortRadius);
mBottomLine.setY(shortRadius);
mPath.reset();
mPath.moveTo(mTopLine.middle.x, mTopLine.middle.y);
mPath.cubicTo(mTopLine.right.x, mTopLine.right.y,
mRightLine.top.x, mRightLine.top.y,
mRightLine.middle.x, mRightLine.middle.y);
mPath.cubicTo(mRightLine.bottom.x, mRightLine.bottom.y,
mBottomLine.right.x, mBottomLine.right.y,
mBottomLine.middle.x, mBottomLine.middle.y);
mPath.cubicTo(mBottomLine.left.x, mBottomLine.left.y,
mLeftLine.bottom.x, mLeftLine.bottom.y,
mLeftLine.middle.x, mLeftLine.middle.y);
mPath.cubicTo(mLeftLine.top.x, mLeftLine.top.y,
mTopLine.left.x, mTopLine.left.y,
mTopLine.middle.x, mTopLine.middle.y);
invalidate();
}
onTouchEvent:
@Override
public boolean onTouchEvent(MotionEvent event) {
switch (event.getAction()) {
case MotionEvent.ACTION_DOWN:
mPrevX = event.getX();
mPrevY = event.getY();
int centerX = getWidth() / 2;
int centerY = getHeight() / 2;
//点击位置x坐标与圆心的x坐标的距离
float distanceX = Math.abs(centerX - mPrevX);
//点击位置y坐标与圆心的y坐标的距离
float distanceY = Math.abs(centerY - mPrevY);
//点击位置与圆心的直线距离
int distanceZ = (int) Math.sqrt(Math.pow(distanceX, 2) + Math.pow(distanceY, 2));
if (distanceZ > radius) {
mIsDown = false;
} else {
mIsDown = true;
return true;
}
break;
case MotionEvent.ACTION_MOVE:
if (mIsDown) {
updatePath(event.getX(), event.getY());
return true;
}
case MotionEvent.ACTION_UP:
case MotionEvent.ACTION_CANCEL:
if (mIsDown) {
float endX = event.getX();
float endY = event.getY();
mIsDown = false;
back(endX, endY); // 还原
return true;
}
break;
}
return super.onTouchEvent(event);
}
// 回去
private void back(final float x, final float y) {
final float diffX = x - mPrevX;
final float diffY = y - mPrevY;
ValueAnimator animator = new ValueAnimator();
animator.setDuration(200);
animator.setFloatValues(0, 1);
animator.setInterpolator(new AccelerateInterpolator());
animator.addUpdateListener(new ValueAnimator.AnimatorUpdateListener() {
@Override
public void onAnimationUpdate(ValueAnimator animation) {
float fract = animation.getAnimatedFraction();
updatePath(x - diffX * fract, y - diffY * fract);
}
});
animator.start();
}
绘制:
private float distance(float x1, float y1, float x2, float y2) {
return (float) Math.sqrt(Math.pow(x1 - x2, 2) + Math.pow(y1 - y2, 2));
}
private static float points2Degrees(float x1, float y1, float x2, float y2) {
double angle = Math.atan2(y2 - y1, x2 - x1);
return (float) Math.toDegrees(angle);
}
@Override
protected void onDraw(Canvas canvas) {
super.onDraw(canvas);
canvas.save();
canvas.translate(getWidth() / 2, getHeight() / 2);
canvas.rotate(mDegree);
canvas.drawPath(mPath, mPaint);
canvas.restore();
}
PathMeasure##
PathMeasure是一个用来测量Path的类,主要有以下方法:
构造方法:
- PathMeasure(): 创建一个空的PathMeasure;
- PathMeasure(path, forceClosed): 创建 PathMeasure 并关联一个指定的Path(Path需要已经创建完成), forceClosed 是否闭合path;
公共方法:
- setPath(Path path, boolean forceClosed) : 关联一个Path;
- isClose(): 是否闭合;
- getLength(): 获取Path的长度;
- nextContour() : 跳转到下一个段(轮廓);
- getSegment(float startD, float stopD, Path dst, boolean startWithMoveTo):
getSegment 用于获取Path的一个片段,将截取成功的路径放入 dst中 - getPosTan(float distance, float[] pos, float[] tan):获取指定长度的位置坐标及该点切线值;
- getMatrix(float distance, Matrix matrix, int flags): 获取指定长度的位置坐标及该点Matrix
getSegment
getSegment 用于获取Path的一个片段,将截取成功的路径放入 dst中;
android4.4或之前,调用此方法,需关闭硬件加速;
startWithMoveTo:如果 startWithMoveTo 为 true, 则被截取出来到Path片段保持原状,如果 startWithMoveTo 为 false,则会将截取出来的 Path 片段的起始点移动到 dst 的最后一个点,以保证 dst 的连续性,也即保证是否形变,或连续;
被截取的片段,会加入到 dst中,而不是覆盖;
nextContour
Path,很有可能是由多个path片段组成,但不论是 getLength , getgetSegment 或者是其它方法,都只会在其中第一条线段上运行,而这个 nextContour 就是用于跳转到下一条曲线到方法,如果跳转成功,则返回 true, 如果跳转失败,则返回 false。
// path 可分段
Path path = new Path();
path.addRect(-100, -100, 100, 100, Path.Direction.CW);
path.addRect(-200, -200, 200, 200, Path.Direction.CW);
path.addCircle(0, 0, 50, Path.Direction.CW);
canvas.drawPath(path, mPaint);
PathMeasure pathMeasure = new PathMeasure(path, false); // 与 path关联
Log.e("TAG", pathMeasure.getLength() + ""); // 800
pathMeasure.nextContour(); // 获取下一个曲线
Log.e("TAG", pathMeasure.getLength() + ""); // 1600
pathMeasure.nextContour();
Log.e("TAG", pathMeasure.getLength() + ""); // 313.6
getPosTan
得到路径上某一长度的位置以及该位置的正切值;
boolean getPosTan (float distance, float[] pos, float[] tan)
distance: 距离Path的起点,0<=distance<=getLength()
pos: 该点的坐标值;
tan: 该点的正切值,是用来判断 Path 的趋势的,即在这个位置上曲线的走向;
如下:
实现代码:
float currentValue = 0.0f;
float[] pos = new float[2]; // 点的时间坐标
float[] tan = new float[2]; // tangent值,用于计算角度
Bitmap mInDicator; // 指示器
Matrix matrix = new Matrix();
// 箭头图片
mInDicator = BitmapFactory.decodeResource(context.getResources(), R.mipmap.indicator);
@Override
protected void onDraw(Canvas canvas) {
super.onDraw(canvas);
// 移动到屏幕中间
canvas.translate(mWidth / 2, mHeight / 2);
// 画圆
Path path = new Path();
path.addCircle(0, 0, 200, Path.Direction.CW);
canvas.drawPath(path, mPaint);
PathMeasure measure = new PathMeasure(path, false);
currentValue += 0.005;
if (currentValue >= 1) {
currentValue = 0;
}
// pos 点的坐标值
// tan 可用来计算点的正切值,是用来判断 Path 的趋势的,即在这个位置上曲线的走向
measure.getPosTan(measure.getLength() * currentValue, pos, tan);
float degree = (float) (Math.atan2(tan[1], tan[0]) * 180 / Math.PI);
/* =============== 第一种方式,转canvas,点为中心选择 ================= */
canvas.save();
canvas.rotate(degree, pos[0],pos[1]);
int halfDrawableWidth = mInDicator.getWidth() / 2;
int halfDrawableHeight = mInDicator.getHeight() / 2;
Rect bounds = new Rect((int) pos[0] - halfDrawableWidth, (int) pos[1] - halfDrawableHeight, (int) pos[0] + halfDrawableWidth, (int) pos[1] + halfDrawableHeight);
canvas.drawBitmap(mInDicator, null, bounds, mPaint);
canvas.restore();
invalidate();
/* =============== 第一种方式,转canvas,点为中心选择 ================= */
/* =============== 第二种方式,利用Matrix选择 ================= */
/*
matrix.reset();
matrix.postRotate(degree, mInDicator.getWidth() / 2, mInDicator.getHeight() / 2); // 旋转图片
// 偏移图片绘制中心,将绘制中心调整到与当前点重合
matrix.postTranslate(pos[0] - mInDicator.getWidth() / 2, pos[1] - mInDicator.getHeight() / 2);
canvas.drawBitmap(mInDicator, matrix, mPaint);
invalidate();
*/
/* =============== 第二种方式,利用Matrix ================= */
/* =============== 第三种方式,利用getMatrix() 简洁 ================= */
matrix.reset();
measure.getMatrix(measure.getLength() * currentValue, matrix, PathMeasure.TANGENT_MATRIX_FLAG | PathMeasure.POSITION_MATRIX_FLAG);
matrix.preTranslate(-mInDicator.getWidth() / 2, -mInDicator.getHeight() / 2);
canvas.drawBitmap(mInDicator, matrix, mPaint);
invalidate();
/* =============== 第三种方式,利用getMatrix() ================= */
getMatrix
得到路径上某一长度的位置以及该位置的正切值的矩阵;上面的第三种方式;