Circle, Sphere: Contact | Fill Level
Contact Sphere - Obstacle Calculation
A calculator for the point where a sphere on a plane - or a circle on a line - touches an obstacle at a certain height. The obstacle must be lower than the sphere radius, otherwise the distance from the contact point of the sphere to the plane is equal to the radius. Please enter two values, the third value and the chord length will be calculated. This calculation applies not only to a sphere, but also to a circle and to a rolling cylinder, for example a tire.
Example: a sphere with a diameter of one meter, so a radius of 50 cm, touches a 10 cm high obstacle. Then the distance of the sphere's center to the obstacle, projected to the plane, is 30 cm.
The calculation is:
Be γ the angle between the two straight lines from the center of the sphere to the touching points.
Be c the chord length between both touching points.
c² = a² + h²
If r and h are given: γ = acos( 1 - h/r )
If r and a are given: γ = asin( a/r )
c = √2 r² * [ 1 - cos(γ) ]
If a and h are given:
γ = 2 acos[ (a² + c² - h²) / (2ac) ]
r = c / sin(γ) * sin[ (π-γ) / 2 ]
The radius is the distance from the center of a sphere or circle to its edge. It is half the diameter, which is the distance between two farthest points on the surface of the sphere or of the circle. The height is the distance, measured vertically upwards, between the ground and the point where the sphere touches the obstacle. If the shape is more complicated than the rectangle in the example above, then this height may be difficult to determine. The same applies to the distance; this is the horizontal length between the point where the sphere touches the ground and the vertical ground line of the obstacle. A chord is simply a straight line between two points on a circle.
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