如图判断点PPP是否在矩形P1P2P3P4P_{1}P_{2}P_{3}P_{4}P1P2P3P4内部?
从上图可以看出:
点 P{P}P 位于矩形内部 ⇔\Leftrightarrow⇔ {∡PP1P4≤90∡PP1P2≤90∡PP3P4≤90∡PP3P2≤90\left\{\begin{aligned} \measuredangle PP_{1}P_{4}\leq90\\ \measuredangle PP_{1}P_{2}\leq90\\ \measuredangle PP_{3}P_{4}\leq90 \\ \measuredangle PP_{3}P_{2}\leq90 \end{aligned}\right.⎩⎪⎪⎪⎪⎨⎪⎪⎪⎪⎧∡PP1P4≤90∡PP1P2≤90∡PP3P4≤90∡PP3P2≤90 ⇔\Leftrightarrow⇔ {P1P⃗⋅P1P4⃗≥0P1P⃗⋅P1P2⃗≥0P3P⃗⋅P3P4⃗≥0P3P⃗⋅P3P2⃗≥0\left\{\begin{aligned} \vec{P_{1}P} \cdot \vec{P_{1}P_{4}} \geq 0\\ \vec{P_{1}P} \cdot \vec{P_{1}P_{2}} \geq 0\\ \vec{P_{3}P} \cdot \vec{P_{3}P_{4}} \geq 0\\ \vec{P_{3}P} \cdot \vec{P_{3}P_{2}} \geq 0\\ \end{aligned}\right.⎩⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎧P1P⋅P1P4≥0P1P⋅P1P2≥0P3P⋅P3P4≥0P3P⋅P3P2≥0
所以,点PPP在矩形P1P2P3P4P_{1}P_{2}P_{3}P_{4}P1P2P3P4内部的条件为:
P1P⃗⋅P1P4⃗≥0\vec{P_{1}P} \cdot \vec{P_{1}P_{4}} \geq 0P1P⋅P1P4≥0 && P1P⃗⋅P1P2⃗≥0\vec{P_{1}P} \cdot \vec{P_{1}P_{2}} \geq 0P1P⋅P1P2≥0 && P3P⃗⋅P3P4⃗≥0\vec{P_{3}P} \cdot \vec{P_{3}P_{4}} \geq 0P3P⋅P3P4≥0 && P3P⃗⋅P3P2⃗≥0\vec{P_{3}P} \cdot \vec{P_{3}P_{2}} \geq 0P3P⋅P3P2≥0
判断点是否在矩形内
最新推荐文章于 2024-04-10 22:39:27 发布