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If k is an irrational number, then the curve never closes, and fills the space between the larger circle and a circle of radius R − 2r. Each hypocycloid (for any value of r) is a brachistochrone for the gravitational potential inside a homogeneous sphere of radius R. [6] The area enclosed by a hypocycloid is given by: [3] [7]
The red curve is an epicycloid traced as the small circle (radius r = 1) rolls around the outside of the large circle (radius R = 3).. In geometry, an epicycloid (also called hypercycloid) [1] is a plane curve produced by tracing the path of a chosen point on the circumference of a circle—called an epicycle—which rolls without slipping around a fixed circle.
Aquatic insects consume the stored air while under water or use it to regulate buoyancy. During a molt, air sacs fill and enlarge as the insect breaks free of the old exoskeleton and expands a new one. Between molts, the air sacs provide room for new growth—shrinking in volume as they are compressed by expansion of internal organs.
The alveoli are tiny air sacs in the lungs where gas exchange takes place. The mean number of alveoli in a human lung is 480 million. [11] When the diaphragm contracts, a negative pressure is generated in the thorax and air rushes in to fill the cavity. When that happens, these sacs fill with air, making the lung expand.
A cycloid generated by a rolling circle. In geometry, a cycloid is the curve traced by a point on a circle as it rolls along a straight line without slipping. A cycloid is a specific form of trochoid and is an example of a roulette, a curve generated by a curve rolling on another curve.
Doing the same at 5500 m, where the atmospheric pressure is only 50 kPa, the intrapulmonary air pressure falls to 25 kPa. Therefore, the same change in lung volume at sea level results in a 50 kPa difference in pressure between the ambient air and the intrapulmonary air, whereas it result in a difference of only 25 kPa at 5500 m.
In the differential geometry of curves, a roulette is a kind of curve, generalizing cycloids, epicycloids, hypocycloids, trochoids, epitrochoids, hypotrochoids, and involutes. On a basic level, it is the path traced by a curve while rolling on another curve without slipping.
Diagram showing displacement of the Sun's image at sunrise and sunset Comparison of inferior and superior mirages due to differing air refractive indices, n. Atmospheric refraction is the deviation of light or other electromagnetic wave from a straight line as it passes through the atmosphere due to the variation in air density as a function of height. [1]