The orange coloured curve is the hyperbola ([math]x^2-y^2=-1[/math]). Triangle PDQ is congruence to triangle PSR with PR equal to OP. Drag Q or P to approach each other and examine the congruence of the green and brown triangles. What is the orientational relationship between the segments OP and PR?
Differentiate the hyperbolic equation [math]x^2-y^2=-1[/math] with respect to [math]\theta[/math] to yield [math]x\frac{dx}{d\theta}-y\frac{dy}{d\theta}=0[/math]. What is the geometric relationship between the vector [math](x,y)[/math] and the gradient [math]\left(\frac{dx}{d\theta},\frac{dy}{d\theta}\right)[/math]? Prove that [math]\left(\frac{dx}{d\theta}\right)^2-\left(\frac{dy}{d\theta}\right)^2=1[/math]. What does the "unity" of the gradient tell us about the parameter [math]\theta[/math]?