The space curve (hodograph) drawn by the point P in space is given by
Next, let's think about what the derivative
of the vector function
represents geometrically.
The derivative of a vector function
,
is by the definition
.
Unfortunately, this limit cannot be used as a tangent direction vector. Because this limit is and
has no direction.
Therefore, to avoid this, consider the following vector that can obtain a large length when is small.
This vector is parallel to
when
is non-zero.That is, this vector is parallel to the tangent direction vector. Therefore, when this limit value
exists, this limit value can be considered as a tangent direction vector, so
is called tangent vector of curve
.
Also,
Note that
here, from the example 2
and
is orthogonal.So,
is called normal vector
. Also, for
,
Answer
The straight line to be found has a starting point of and a direction of
. Let the arbitrary point on the line as
. Then,
Answer
Since
, we have
.t should be noted here that the value of
is from
to
, so the curve to be calculated is
.
Answer The tangent vector is give by
When
satisfies the condition
,we say
is smooth curve.The length of the
part of a smooth curve
is called arc length an denoted by
.
How can you find the arc length?
Curve
corresponding to the small interval
in the interval
. The length of the arc PQ is considered to be approximated by the line segment PQ. Then
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As you may have noticed here, if you think of as time, then
is considered to be a change in position within a minute time. Therefore, it represents the moving speed of the point P. Therefore, the length
of the curve can be thought of as the distance that the point P has moved within the time
.
Answer
Since
,we have
. Thus,
is a smooth curve that spirally rotates around a cylinder with a radius of 1.
Answer
implies
When the curve is expressed by
using the parameter
,we find the tangent vector.
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Now let's think about how the curve bends. First, let's think about the curve on the plane.Let be the angle formed by the tangent
and the
axis at the point P on the curve. The tangents
and
change as the point P moves.At this time, the rate of change of
per unit arc length is called curvature(curvature) .
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Where the tangent unit vector can be represented by
.Therefore, when examining the rate of change of the tangent vector per unit arc length,
Answer
First find
.
is the slope of the tangent line. Thus
Therefore,
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The curvature of the curve in space is
.