曲线的切线与法平面
曲线 Γ : { x = φ ( t ) y = ψ ( t ) z = ω ( t ) \Gamma:\begin{cases}x=\varphi(t) \\ y=\psi(t) \\ z=\omega(t) \end{cases} Γ:⎩⎪⎨⎪⎧x=φ(t)y=ψ(t)z=ω(t)， t ∈ [ α , β ] t\in[\alpha,\beta] t∈[α,β]
切线方程： x − x 0 φ ′ ( t 0 ) = y − y 0 ψ ′ ( t 0 ) = z − z 0 ω ′ ( t 0 ) \frac{x-x_{0}}{\varphi'(t_{0})}=\frac{y-y_{0}}{\psi'(t_{0})}=\frac{z-z_{0}}{\omega'(t_{0})} φ′(t0)x−x0=ψ′(t0)y−y0=ω′(t0)z−z0
法平面方程： φ ′ ( t 0 ) ( x − x 0 ) + ψ ′ ( t 0 ) ( y − y 0 ) + ω ′ ( t 0 ) ( z − z 0 ) = 0 \varphi'(t_{0})(x-x_{0})+\psi'(t_{0})(y-y_{0})+\omega'(t_{0})(z-z_{0})=0 φ′(t0)(x−x0)+ψ′(t0)(y−y0)+ω′(t0)(z−z0)=0
曲面的切平面与法线
曲面 Σ : x = F ( x , y , z ) = 0 \Sigma:x=F(x,y,z)=0 Σ:x=F(x,y,z)=0
切平面方程： F x ( x 0 , y 0 , z 0 ) ( x − x 0 ) + F y ( x 0 , y 0 , z 0 ) ( y − y 0 ) + F z ( x 0 , y 0 , z 0 ) ( z − z 0 ) = 0 F_{x}(x_{0},y_{0},z_{0})(x-x_{0})+F_{y}(x_{0},y_{0},z_{0})(y-y_{0})+F_{z}(x_{0},y_{0},z_{0})(z-z_{0})=0 Fx(x0,y0,z0)(x−x0)+Fy(x0,y0,z0)(y−y0)+Fz(x0,y0,z0)(z−z0)=0
法线方程： x − x 0 F x ( x 0 , y 0 , z 0 ) = y − y 0 F y ( x 0 , y 0 , z 0 ) = z − z 0 F z ( x 0 , y 0 , z 0 ) \frac{x-x_{0}}{F_{x}(x_{0},y_{0},z_{0})}=\frac{y-y_{0}}{F_{y}(x_{0},y_{0},z_{0})}=\frac{z-z_{0}}{F_{z}(x_{0},y_{0},z_{0})} Fx(x0,y0,z0)x−x0=Fy(x0,y0,z0)y−y0=Fz(x0,y0,z0)z−z0