- #1
dispiriton
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Homework Statement
In general how do i parametrize a circle of radius r at centre (a,b,c) laying on a plane? E.g. (x + y + z = 6)
In R3, circles can be parameterized using the following equations:
x = cx + rcos(t)
y = cy + rsin(t)
z = cz
where (cx, cy, cz) is the center of the circle and r is the radius. The parameter t ranges from 0 to 2π.
No, circles in R3 can only be parameterized if they lie in a plane. This is because the equations used to parameterize circles in R3 assume that the z-coordinate remains constant, which is only true for circles lying in a plane.
Parameterizing circles in R3 allows for a systematic way of representing and manipulating circles in three-dimensional space. It also simplifies calculations involving circles, such as finding intersections with other objects or calculating the length of an arc.
Yes, there is a different method for parameterizing circles in R3 with a non-zero z-coordinate. It involves using the equations:
x = cx + rcos(t)
y = cy + rsin(t)
z = cz + h
where (cx, cy, cz) is the center of the circle, r is the radius, and h is the desired height of the circle above the xy-plane.
The parameterization of circles in R3 can be used in various real-world applications, such as computer graphics, physics, and engineering. For example, in computer graphics, circles can be represented and manipulated using their parameterization, allowing for the creation of 3D images and animations. In physics and engineering, the parameterization of circles can be used to model and analyze circular motion in three-dimensional space.