Lecture 2 Differential geometry of curves 2.1 Definition of curves 2.1.1 Plane curves Implicit curves f(, y)=0 Example:x2+y2=a2 It is difficult to trace implicit curves It is easy to check if a point lies on the curve Multi-valued and closed curves can be represented It is easy to evaluate tangent line to the curve when the curve has a vertical or near vertical tangent Axis dependent(Difficult to transform to another coordinate system) Example:x3+y=3xy: Folium of Descartes(see Figure 2.1a) Let f(e, y= +y-3.ry=0, f(0,0)=0=(a, y)=(0, 0) lies on the curve Example: If we translate by(1, 2)and rotate the axes by 0=atan(3), the hyperbola x-5=l, shown in Figure 2. 1(b), will become 2x--72ry+23y-+140.T-20y+50=0 Explicit curves y= f(a) One of the variables is expressed in terms of the other It is easy to trace explicit curves It is easy to check if a point lies on the curve Multi-valued and closed curves can not be easily represented It is difficult to evaluate tangent line to the curve when the curve has a vertical or near vertical tangentLecture 2 Differential geometry of curves 2.1 Definition of curves 2.1.1 Plane curves • Implicit curves f(x, y) = 0 Example: x 2 + y 2 = a 2 – It is difficult to trace implicit curves. – It is easy to check if a point lies on the curve. – Multi-valued and closed curves can be represented. – It is easy to evaluate tangent line to the curve when the curve has a vertical or near vertical tangent. – Axis dependent. (Difficult to transform to another coordinate system). Example: x 3 + y 3 = 3xy : Folium of Descartes (see Figure 2.1a) Let f(x, y) = x 3 + y 3 − 3xy = 0, f(0, 0) = 0 ⇒ (x, y) = (0, 0) lies on the curve Example: If we translate by (1,2) and rotate the axes by θ = atan( 3 4 ), the hyperbola x 2 4 − y 2 2 = 1, shown in Figure 2.1(b), will become 2x 2 −72xy+23y 2 +140x−20y+50 = 0. • Explicit curves y = f(x) One of the variables is expressed in terms of the other. Example: y = x 2 – It is easy to trace explicit curves. – It is easy to check if a point lies on the curve. – Multi-valued and closed curves can not be easily represented. – It is difficult to evaluate tangent line to the curve when the curve has a vertical or near vertical tangent. 2