Curves in differential geometry
Encyclopedia
This page addresses the mathematical topic of curve
s in differential geometry.
defines the constant curve, a parametric curve of class C∞. The image of the constant curve is the single point p. The curve is closed and analytic but not simple.
. A parametric definition of a line through the points p0 and p1 (p0 ≠ p1 and p0,p1 ∈ R3) is given by
The image of the curve is a line. Note that
is a different curve but the image of both curves is the same line.
defines a helix
circling the z-axis.
Curve
In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...
s in differential geometry.
Constant curve
Given a point p0 in R3 and a subinterval I of the real line,defines the constant curve, a parametric curve of class C∞. The image of the constant curve is the single point p. The curve is closed and analytic but not simple.
Line
A slightly more complex example is the lineLine (mathematics)
The notion of line or straight line was introduced by the ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects...
. A parametric definition of a line through the points p0 and p1 (p0 ≠ p1 and p0,p1 ∈ R3) is given by
The image of the curve is a line. Note that
is a different curve but the image of both curves is the same line.
Helix
Given r, ω in Rdefines a helix
Helix
A helix is a type of smooth space curve, i.e. a curve in three-dimensional space. It has the property that the tangent line at any point makes a constant angle with a fixed line called the axis. Examples of helixes are coil springs and the handrails of spiral staircases. A "filled-in" helix – for...
circling the z-axis.