Why Small Circles Have Large Curvature

A smaller circle has a larger curvature

Compare the cyan circle with the fixed gold reference. Both currently have radius one half, so their curvature magnitudes are both two. Curvature measures how tightly a circle turns: for a circle, its magnitude is one divided by its radius.

Make the cyan circle smaller, with radius one quarter. Its curvature is now four. The radius was divided by two while the curvature was multiplied by two. These measurements move in opposite directions. A larger curvature therefore describes a tighter circle, not a larger circle.

Return to radius one half. The curvature returns to two, and the cyan circle matches the gold reference again. The two circles touch externally at one point. Their touching relationship remains intact while we change the cyan radius and move its center accordingly.

Now enlarge the cyan radius to one. Its curvature becomes one. The new radius is twice the reference radius, and its curvature is half the reference curvature. Radius and curvature happen to have equal numerical values at one, because one is its own reciprocal.

Read exact fractions and decimal measurements

Set the radius to three quarters. The curvature is four thirds, which the decimal readout approximates as one point three three three. The reciprocal relation applies to fractions as well as whole numbers. We have not rounded the mathematical definition merely because the screen has limited digits.

Compare radius two fifths, or zero point four. Its reciprocal is five halves, or two point five. Multiply the radius and curvature: two fifths times five halves equals one. This product is a useful check that the two readouts describe the same circle.

Set the radius to one fifth. Its curvature is five. The smaller radius makes a tighter turn, and the reciprocal value gets larger. Predicting the direction of change before doing the arithmetic helps prevent confusing curvature with circumference, diameter, or area.

Return to radius one half for a familiar comparison. Diameter would be twice the radius, while curvature is its reciprocal. These are different operations. For this circle, the diameter is one and the curvature is two. Neither value is another name for its radius.

Touching is a geometric condition

Approach the contact between the two circles. Their boundaries meet at one point without crossing. This is external tangency. The line joining their centers passes through that contact, and the distance between centers equals the sum of their positive radii.

Compare a smaller cyan circle at the same contact. Its radius is one quarter and the fixed reference radius is one half. Their center distance is therefore three quarters. The contact stays fixed while the cyan center moves closer to it.

Compare a cyan radius of three quarters. The center distance is now one and one quarter: one half plus three quarters. Keeping the centers fixed while increasing the radius would generally make the boundaries cross. The native comparison moves the center to preserve tangency.

Pull back to see the complete pair. The guide shows a radius, not a diameter. Its endpoints are the circle’s center and one point on its boundary. That visible segment gives the geometric meaning of the radius used in the reciprocal calculation.

An enclosing orientation receives a minus sign

Switch to the internally tangent comparison. The gold enclosure now has radius one, and the cyan circle has radius one half. Both physical radii are positive. In the signed-curvature convention for circle packings, the enclosure receives curvature minus one, while the inner circle receives positive two.

Reduce the inner radius to one quarter. Its positive curvature becomes four. The center distance to the unit enclosure’s center is three quarters: the enclosing radius minus the inner radius. Internal tangency uses a difference of radii, rather than their sum.

Approach the internal contact. The negative sign assigned to the enclosure records orientation within the configuration. It does not say that the gold circle has a negative physical radius or that a distance is negative. Curvature magnitude and signed curvature must be distinguished.

Pull back and compare an inner radius of three quarters. Its curvature magnitude is four thirds, and the center separation is one quarter. As the inner circle grows, that center moves toward the enclosure’s center so the boundaries continue to touch internally.

Changing size and changing the camera are different

Return to the external comparison with cyan radius one half and curvature two. Imagine enlarging this mathematical radius by a factor of three. The new radius would be one and a half, and its curvature would be two thirds. We will show that radius next.

Here is radius one and a half. The curvature readout is approximately zero point six six seven. Multiplying a positive radius by three divides its reciprocal by three. The gold reference was held fixed so the size change remains perceptible in the comparison.

Move closer to the contact while keeping that radius unchanged. The circle looks larger on screen, but its radius and curvature readouts remain one and a half and two thirds. A camera approach changes pixels per unit. It does not change the mathematical circle.

Pull back to the whole comparison. If a radius is measured in centimeters, curvature is measured per centimeter. Changing the length unit changes the numerical curvature. An integer curvature claim therefore belongs to a specified scale and unit; it is not an unqualified statement about appearance.

A large circle looks flatter near contact

Increase the cyan radius to two. Its curvature is one half. Near the fixed tangency point, the visible arc bends less tightly than the reference circle. A large circle can extend beyond the viewport while its local arc still makes this comparison clear.

Compare radius ten and curvature one tenth. We are inspecting a small part of a much larger circle, not claiming that its entire boundary fits on screen. The arc near contact is flatter. As positive radius grows without bound, curvature approaches zero.

The straight line is a zero-curvature limiting case, not an ordinary circle with a finite radius. This display still uses radius ten. A nearly straight-looking arc remains curved. The limiting statement describes what happens as radius increases indefinitely, beyond any one displayed value.

Compare the opposite direction: radius one tenth and curvature ten. The circle is small and turns tightly. As a positive radius approaches zero, reciprocal curvature grows without bound. At radius zero, division by zero does not give a finite curvature for a regular circle.

Check a reciprocal comparison without rounding the geometry

Use a cyan radius of one third next to the fixed reference radius one half. Its curvature is three, compared with reference curvature two. The cyan circle has two thirds of the reference radius but three halves of its curvature. A ratio reverses when we take reciprocals.

Approach their external contact. The center distance is one half plus one third, or five sixths. That sum uses positive physical radii. Curvature is useful for other calculations, but adding three and two would not give this center distance. Identify which geometric quantity a formula is measuring before substituting its numbers.

Double the cyan radius to two thirds. Its curvature halves to three halves, and its center moves to preserve the same tangency point. The fixed reference does not change. We have one controlled comparison: changing one radius while keeping the external touching condition and reference circle intact.

At radius two thirds, the center distance is seven sixths, larger than one. The circles still touch externally because the center distance equals the sum of their radii. A contact condition does not require equal circles, equal curvatures, or a center distance of one in our chosen units.

Return to equal radii one half. The curvature values agree again and the center distance is one. Use this symmetric case as a check, then compare it with the unequal pair. Reciprocal radius, sum of radii, and equality of radii are three distinct operations visible in the same simple diagram.

Carry the measurements into tangent packings

Return to the internally tangent half-radius example. The signed values minus one for the enclosure and positive two for the inner circle will recur in the next lesson. These values encode reciprocal radii together with the chosen enclosure orientation.

Show the planar packing with its centers and contact guides. Several circles now constrain one another at once. Their curvatures remain reciprocal radii, but tangency imposes additional relations among those measurements. The next lesson will determine a fourth circle from three fixed tangent neighbors.

Read the large inner circles and their smaller neighbors. Large curvature labels belong to small circles. A label is a measurement of the corresponding circle, not a count of how many circles touch it. The center and contact guides describe different information about the same configuration.

Return to the complete configuration. Curvature magnitude is reciprocal radius; signed curvature also records orientation. Tangency constrains centers and radii together, and camera motion leaves those mathematical values unchanged. You can now read the measurements that the packing equations will connect.