PROJECTION OF LINE-SEGMENTS

Excerpts

• PROJECTION OF LINE-SEGMENTS

  The length of the projection of a line-segment upon a given line is equal to the length of the line-segment multiplied by the cosine of the projection angle.

• PROJECTION OF LINE-SEGMENTS

  Likewise we define the *sine* of an acute angle of a right triangle as the *ratio of the opposite side to the hypotenuse*, and the *tangent* of an acute angle of a right triangle as the *ratio of the opposite side to the adjacent side*.

• PROJECTION OF LINE-SEGMENTS

  The area of the projection of a plane-segment on a plane is equal to the area of the plane-segment multiplied by the cosine of the projection angle.

• PROJECTION OF LINE-SEGMENTS

  The altitude of an oblique prism or cylinder is equal to an element multiplied by the cosine of the angle between the plane of the base and that of a right section.

• PROJECTION OF LINE-SEGMENTS

  Note that when $a$ and $b$ are equal, the ellipse becomes a circle, and this formula reduces to $\pi a^2$ as it should.

• PROJECTION OF LINE-SEGMENTS

  The comparatively low temperature of the earth’s surface near the pole, even in summer, when the sun does not set for months, is due partly to the *obliqueness* with which the sun’s rays strike the earth.

• PROJECTION OF LINE-SEGMENTS

  We assume that it lies halfway between these numbers. This assumption, while not quite correct, is very nearly so for small differences of angles, as in this case, where the total difference is only one degree.

Equations

• PROJECTION OF LINE-SEGMENTS

  \dfrac{p}{l} = \text{cosine}\, \angle BAE

  The cosine of the projection angle is the ratio of the length p of a segment's projection to the length l of the segment.

• PROJECTION OF LINE-SEGMENTS

  \sin A = \dfrac{a}{c},\quad \cos A = \dfrac{b}{c},\quad \tan A = \dfrac{a}{b}

  In a right triangle, the sine of an acute angle A is opposite side over hypotenuse, its cosine is adjacent side over hypotenuse, and its tangent is opposite side over adjacent side.

• PROJECTION OF LINE-SEGMENTS

  p = l \cos A

  The length of the projection of a line-segment on a given line equals the length of the segment multiplied by the cosine of the projection angle.

• PROJECTION OF LINE-SEGMENTS

  BE &= AB \cdot \cos D

  The altitude of an oblique prism or cylinder equals an element multiplied by the cosine of the dihedral angle between the base plane and the right-section plane.

• PROJECTION OF LINE-SEGMENTS

  c &= b \cos \angle 1

  The area of the projection of a plane-segment equals the area of the plane-segment multiplied by the cosine of the projection angle.

• PROJECTION OF LINE-SEGMENTS

  S' &= S \cos \angle 1

  For a rectangle with one side parallel to the line of intersection of the planes, its projection has area equal to the rectangle's area times the cosine of the projection angle.

• PROJECTION OF LINE-SEGMENTS

  \pi ab

  The area of an ellipse, formed as the projection of a circle, equals pi times the semimajor axis times the semiminor axis.

• PROJECTION OF LINE-SEGMENTS

  ABCD = A'BCD' \cos \angle 1

  The right cross-section of a beam of sunlight equals the area it spreads over on the ground multiplied by the cosine of the angle between the rays and the ground.

Problems

No exercises in this chapter.