1.3.6 Hyperbolic Curvature

The recent sec­tions re­volved around cur­va­ture for the circle in the plane, and in this sec­tion, we pivot to con­sid­er the hy­per­bol­ic curve in the plane. Cir­cu­lar dy­nam­ics are in­her­ent in many nat­ur­al phe­nom­e­na, par­tic­u­lar­ly in the study of wave me­chan­ics. Like the circle, the hy­per­bo­la also plays a cen­tral role in the phe­nom­e­nol­o­gy sur­round­ing us. This is par­tic­u­lar­ly true re­gard­ing the shape of open orbits in space­time and, most no­tably, the geom­e­try of spe­cial rel­a­tiv­i­ty. The gen­er­al equa­tion for the hy­per­bo­la that opens to the left and right, is typ­i­cal­ly pre­sent­ed in the form

for the point $(h,k)$, rep­re­sent­ing the center of the hy­per­bo­la. The term $a$ rep­re­sents the semi-major axis, which is the dis­tance from the hy­per­bo­la's center to the pos­i­tive and neg­a­tive ver­tices $\left( V_{1},V_{2} \right),$where the hy­per­bo­la cross­es the hor­i­zon­tal $x$-axis. While the $b$-term will scale the ver­ti­cal com­po­nent values of the hy­per­bo­la, which can change the ec­cen­tric­i­ty of the curve in the plane. The foci $\left( F_{1},F_{2} \right)$ of the hy­per­bo­la are lo­cat­ed at a dis­tance $c$ from the center of the hy­per­bo­la in the pos­i­tive and neg­a­tive di­rec­tions, and the Pythagore­an iden­ti­ty de­ter­mines the mag­ni­tude of this dis­tance

The Eu­clid­ean plane de­fines the hy­per­bo­la as a set of points with a semi-major axis $a > 0$, as the ab­solute dif­fer­ence be­tween the fixed foci $F_{1},F_{2}$ to the point $\alpha$ on the hy­per­bo­la. The set of points is there­fore, de­fined in set no­ta­tion, as

where the in­vari­ant dis­tance for all points $\alpha$ on the hy­per­bo­la is equal to $2a$, re­ferred to as the major axis. In Figure 1.3.14, we plot a hy­per­bo­la cen­tered at the origin, de­fined by the ex­pres­sion

where, in this case, $a = b$. Like the twice dif­fer­en­tiable equa­tion for the unit circle, we can rep­re­sent the unit hy­per­bo­la in the plane with the in­vari­ant ex­pres­sion

The unit hy­per­bo­la's curves are cen­tered at the origin and ori­ent­ed along the hor­i­zon­tal $x$-axis, with semi-major axis values $a = 1$, ver­tices $V_{1}$ and $V_{2}$, lo­cat­ed at the co­or­di­nates $V_{1} = (1,0),$ $V_{2} = ( - 1,0),$and foci lo­cat­ed at $F_{1} = \left( \sqrt{2},0 \right),F_{2} = \left( - \sqrt{2},0 \right).$In the Eu­clid­ean plane, the unit hy­per­bo­la rep­re­sents the plot seen in the Figure 1.3.15.

ver­tices $V_{1} = (1,0),\ V_{2} = ( - 1,0)$, and as­ymp­totes in grey $(y = x),(y = - x).$

The cur­va­ture and signed cur­va­ture for­mu­las de­rived in Sec­tion 1.3.4, also hold for the hy­per­bo­la in the plane. Re­call­ing the signed cur­va­ture for­mu­la

and its para­met­ric form

To begin, we con­sid­er the cur­va­ture of the hy­per­bo­la in the plane by im­plic­it dif­fer­en­ti­a­tion its in­vari­ant form

the solve for $\frac{dx}{dy}$

re­call­ing that $x^{'} = \frac{dx}{dy}.$Next, we cal­cu­late the second-order de­riv­a­tive $x''$, to find

Taking the $x^{'},x''$de­riv­a­tives, we can now cal­cu­late the cur­va­ture of our unit hy­per­bo­la to be

By defin­ing $\rho = \sqrt{x^{2} + y^{2}}$, to rep­re­sent the Eu­clid­ean dis­tance from the center of the hy­per­bo­la to a point $\alpha$ on the curve, the signed cur­va­ture equals

As an ex­am­ple, we con­sid­er the point $\alpha = \left( 2,\sqrt{3} \right)$ on the unit hy­per­bo­la, find­ing $\rho$ equals

and thus, the signed cur­va­ture will equal

In Figure 1.3.16, the point $\alpha = \left( 2,\sqrt{3} \right)$ is seen on the unit hy­per­bo­la, and the $\rho$-seg­ment rep­re­sents the hy­potenuse of the tri­an­gu­lar poly­gon de­pict­ed in red.

The radius of cur­va­ture of the cir­cu­lar arc at a given point on the unit hy­per­bo­la, is

and for a de­fined point, $\alpha = \left( 2,\sqrt{3} \right)$, $R$ then equals

The radius of cur­va­ture vector $R$, is tan­gent to the unit hy­per­bo­la at this point, and the cir­cu­lar arc will have its center $O'$, lo­cat­ed at the point

of cur­va­ture $R = \rho^{3} = 7\sqrt{7}$, tan­gent to the unit hy­per­bo­la at the point, $\alpha.$

Figure 1.3.17 demon­strates the radius of cur­va­ture$\ R,$at the point $\alpha = \left( 2,\sqrt{3} \right)$, where a cir­cu­lar arc is shown with ar­clength

Next, we con­sid­er the pa­ra­me­ter­ized form for the unit hy­per­bo­la in the plane,

and using the pa­ra­me­ter­ized signed cur­va­ture for­mu­la

we eval­u­ate, to find

Given that $\rho$ is the mag­ni­tude of the seg­ment from the origin to the point lo­cat­ed at $(\sinh(t),\cosh(t))$,

the signed cur­va­ture $k$, be­comes

which is the same result found for the non-pa­ram­e­trized form. Given that

we can ex­press the signed cur­va­ture for the unit hy­per­bo­la, as

The high­est mag­ni­tude of signed cur­va­ture for the unit hy­per­bo­la occurs for $t = 0$

and for the limit, as $t$ ap­proach­es in­fin­i­ty, equals the zero cur­va­ture of flat space

and using the hy­potenuse $\rho$, the fol­low­ing limit also ap­proach­es in­fin­i­ty,

A par­al­lel exists be­tween the cur­va­tures of the hy­per­bo­la and the circle in the plane, where each curve will flat­ten with in­creas­ing values for their hy­potenuse com­po­nents $r,\ \rho$. As $r,\rho \rightarrow \infty$, both curves become straight lines, de­fined with zero cur­va­ture

for which, the unit hy­per­bo­la will flat­ten much faster given the more sig­nif­i­cant degree of the $\rho^{3}$ term in its limit. In the fol­low­ing figure, we plot the func­tions for the signed cur­va­tures of unit hy­per­bo­la $k = \rho^{- 3}$, and the circle of radius $r$ in the plane $\kappa = r^{- 1}$, where we assume $r \in \mathbb{R}^{+}$.

The plots show that the unit hy­per­bo­la's cur­va­ture quick­ly ap­proach­es the hor­i­zon­tal axis's zero cur­va­ture. In con­trast, the cur­va­ture of the circle ap­proach­es the hor­i­zon­tal axis more slowly as we move away from the origin. As the radius of the circle de­creas­es, the cur­va­ture in­creas­es and will ap­proach an in­fi­nite cur­va­ture as the radius ap­proach­es zero

It is im­por­tant to note that the hy­per­bo­la can also be scaled by a scalar radial com­po­nent value, seen by adding the radial com­po­nent $r$ to the pa­ram­e­trized func­tion

and eval­u­at­ing for the signed cur­va­ture

find­ing that the hy­per­bo­la's cur­va­ture also takes on the in­verse radial char­ac­ter­is­tic like the circle.

of the hy­per­bo­la is seen in blue, $k = \frac{1}{{r\rho}^{3}}$, and the cur­va­ture of the circle is seen in red, $k = \frac{1}{r}.$

Com­par­ing the plots for the cur­va­ture of the hy­per­bo­la and circle in Figure 1.3.19, shows that both will ap­proach in­fin­i­ty for small­er scalar radial values of$\ r$, but given the larger degree of the hy­per­bo­la's $\rho^{3}$-term, its cur­va­ture func­tion will ap­proach in­fin­i­ty at a slower rate

This leads us to con­clude that a more ac­cu­rate way to con­sid­er cur­va­ture for hy­per­bol­ic and cir­cu­lar curves in the plane is to com­pare their unit rep­re­sen­ta­tions and, re­spec­tive­ly, the ra­di­al­ly scaled curves from Figure 1.3.19. For the case of the unit curves, the circle's cur­va­ture mag­ni­tude does not change, $\kappa = 1$, while the unit hy­per­bo­la's cur­va­ture mag­ni­tude may con­sid­er­ably change, given the power in the de­nom­i­na­tor of $\kappa = \frac{1}{\rho^{- 3}}.$