KIYA chahte hain:θ line (dir b) aur plane (normal n) ke beech.
Step 1 — Normal ko reference ki tarah use karo.
Normal poori plane ke perpendicular hoti hai. b aur n ke beech ka angle γ dot product se nikalta hai:
cosγ=∣b∣∣n∣b⋅n.Yeh step kyun? Dot product hi hamara ek ऐसा tool hai jo do directions ko angle mein convert karta hai.
Step 2 — γ aur θ ko relate karo.
Line, plane par uski shadow, aur normal ko picture karo. Normal shadow ke 90∘ par khadi hai. Toh line, normal aur shadow ek right triangle banate hain.
θ+γ=90∘⇒γ=90∘−θ.Yeh step kyun? Line-to-plane aur line-to-normal geometry se complementary hote hain.
Absolute value kyun? Line direction b kisi bhi taraf point kar sakta hai, aur θ ko [0∘,90∘] mein rehna chahiye, toh hum sinθ≥0 rakhne ke liye magnitude lete hain.
Socho dhoop seedhi neeche pad rahi hai ek tedhi chhadi par jo zameen se bahar nikal rahi hai. Chhadi line hai, zameen plane hai. Chhadi aur zameen par uski shadow ke beech ka angle "line–plane angle" hai. Ab ek flagpole bhi hai jo bilkul seedha khada hai (normal). Chhadi pole se kuch dur jhuki hui hai. Kyunki pole aur zameen ek perfect right angle banate hain, chhadi jitna angle pole ke saath banati hai, zameen ke saath "bacha hua" angle banati hai. Dono milake hamesha right angle dete hain. Angle ka woh "bacha hua" hona hi exactly woh reason hai jis wajah se hum cos ki jagah sin use karte hain.
Formula for angle between line (dir b) and plane (normal n)
sinθ=∣b∣∣n∣∣b⋅n∣
Line-and-plane mein sin kyun, cos kyun nahi?
Plane ko uske normal se represent kiya jaata hai, jo plane se 90∘ off hai, toh cosγ=cos(90−θ)=sinθ.
Line ke plane ke parallel hone ki condition
b⋅n=0 (direction normal ke perpendicular).
Line ke plane mein lie karne ki condition
b⋅n=0 AUR line ka ek point plane equation satisfy kare.
Line ke plane ke perpendicular hone ki condition
b∥n (toh sinθ=1,θ=90∘).
Line–plane angle θ ki range
0∘≤θ≤90∘.
Formula mein absolute value kyun lete hain?
Line direction unsigned hota hai; ∣⋅∣ se sinθ∈[0,1] rehta hai.
