5.6: Lengths And Areas Given Intrinsically

In our study of surfaces we categorize various properties as either intrinsic or extrinsic. A property is said to be intrinsic if it can be expressed only in terms of the metric coefficients. If this is not the case, then it is said to be extrinsic. If an object is not intrinsic, one needs a parametrization of the surface to study it. Many interesting problems in the theory of surfaces arise in determining whether or not a given property is intrinsic.

Given the metric coefficients g_ij(u,v) , it is possible to find lengths of curves on the domain even if we do not know the definition of a surface in three-space that determines the coefficients. For example, suppose we are given g₁₁(u,v)= 1/v² , g₁₂(u,v)=0 , and g₂₂(u,v)= 1/v² , where u can be any real number and v is strictly positive. Then g₁₁(u,v)g₂₂(u,v)-g₁₂( u,v)²=1/v ⁴ , and since this is never zero, it is possible for these to be the metric coefficients of a regular surface. This g that we have obtained can be used as a metric.

The total length is printed in the Length of Curves on Surfaces control panel window. The arclength function is drawn in the window that has that name.

This demo looks at curves in the upper half plane with the metric given in the previous paragraph, i.e g₁₁=g₂₂ =1/v² and g₁₂=g₂₁ =0.

Find the lengths of a number of curves joining the points (-1,1) and (1,1) in the domain. One is the straight line (u(t),v( t))=(t,1 ) as t goes from -1 to 1 . Another is an arc of the circle centered at the origin with radius 2 . Show that, using this metric, the straight line segment is a longer curve than the arc of circle. What will the "shortest" curve be joining these two points?

Since the integrand for the area of a region of a surface is expressed in terms of the metric coefficients of that surface, the same is true of our formula to compute the area of rectangles on the surface. The metric coefficients completely determine the effects a parametrization of the surface would have on the parameter domain.

To select the rectangle in the domain we just change the domain boundaries and observe the changes. The total area of the domain chosen using our hyperbolic metric is displayed in the control panel under Area of Surface .

This demo shows the domain and area element graph for a region in the upper half plane, using the hyperbolic metric.

Culculate the area of rectangles in the hyperbolic upper half-plane using the previous expression for g.