Today is our topic of discussion Some Elementary Definition .
Some Elementary Definition
Drawing a three-dimensional figure on a two-dimensional page or board is a bit complex. So, drawing pictures with definitions in the classroom will help the students grasp the concepts easily.
1. Plane surface:
If the straight line joining any two points on a surface lies totally on that surface, then the surface is called a plane surface or simply a plane. The upper surface of the still water of a pond is a plane. The smooth floor of a room polished with cement or mosaic is considered to be a plane. But geometrically it is not a plane, for there are high and low points on the floor. In the above picture, ABCD, ADEF, ABGF all are planes.
Observation:
Unless otherwise mentioned, lines and planes in solid geometry are regarded as infinitely extended. Hence it may be inferred from the definition of a plane that if one part of a straight line lies in a plane then the other part cannot be outside it.
2. Curved surface:
If the straight line joining any two points on a surface does not lie wholly in the surface, then the surface is called a curved surface. The surface of a sphere is a curved surface.
3. Solid geometry:
The branch of mathematics which concerns with the properties of solids and surfaces, lines and points is called solid geometry. Sometimes it is called Geometry of Space or Geometry of Three Dimensions.
4. Coplanar straight lines:
If two or more straight lines lie in the same plane or a plane can be made to pass through them, then these straight lines are said to be coplanar. In the figure above AB and CD are coplanar straight lines but EF is not coplanar with them.
5. Skew or non coplanar lines:
Straight lines which do not lie in one plane or through which a plane cannot be made to pass are called skew or non- coplanar straight lines. In the figure above, AB and EF are skew lines. If two pencils are tied cross-wise like a plus sign or a multiplication sign, two non-coplanar lines are formed.
6. Parallel Straight lines:
Two coplanar straight lines are said to be parallel, when they do not intersect each other, i.e., they have no common point. In the figure above, AB and CD are parallel straight lines.
7. Parallel planes or surfaces:
Two planes are said to be parallel when they do not intersect, that is, they do not have any common point. In the figure above ABCD and EFGH are parallel planes.
8. Line parallel to a plane:
If a plane and a straight line are such that they do not intersect though they are extended indefinitely, then the straight line is said to be parallel to the plane. In the figure above, CD is parallel to plane ABGF.
9. Normal or perpendicular to a plane:
A straight line is said to be normal or perpendicular to a plane when it is perpendicular to every straight line in the plane meets it. In the figure below on left, OP is normal to the plane, as it is normal to all of AB, CD, EF residing on the plane.
10. Oblique Line:
A straight line is said to be an oblique line to a plane if it is neither parallel nor perpendicular to the plane. In the figure above on right, MN, ST are oblique lines.
11. Vertical Line of Plane:
A straight line or a plane is said to be vertical when it is parallel to plumb line hanging freely at rest. In the figure below on left, ABCD is a vertical plane and PR vertical line.
12. Horizontal line or plane:
A plane is said to be horizontal when it is perpendicular to a vertical line. Again a straight line is said to be horizontal when it is perpendicular to a vertical line or when it lies in a horizontal plane. In the figure above on left, ABEF is a horizontal plane and PQ is a horizontal line.
13. Planar and skew quadrilateral:
A quadrilateral is said to be plane when its sides lie in the same plane. Again a quadrilateral whose sides do not lie in the same plane is called skew quadrilateral. Two adjacent sides of a skew quadrilateral lie in one plane and the other two adjacent sides lie in another plane. Hence the opposite sides of a skew quadrilateral are also skew. In the above figure on right, ABEF is a planar quadrilateral and BCFE is a skew quadrilateral.
14. Angle between two skew straight lines:
The angle between two skew straight lines is the angle between one of them and the line drawn through any point in that line parallel to other. Again if two straight lines parallel to skew straight lines are drawn at a point, then the angle formed at that point is equal to the angle between the skew straight lines.
Let AB and CD be the skew lines. Take any point O and through O, draw OP, OQ, parallel to AB, CD respectively. Then the angle POQ indicates the angle between the skew lines AB and CD. In other words, ZBRQ also denotes the angle between AB and CD where R is on AB and QR is parallel to CD.
15. Dihedral angle:
If two planes intersect in a straight line, then the angle between the two straight lines drawn from any point on the line of intersection and at right angles to the intersection line is called a dihedral angle
The two planes AB and CD intersect along the straight line AC. From O, any point on AC, two straight lines OM in the plane AB and ON in the plane CD are drawn such that each is perpendicular to AC at O. Then ZMON is the dihedral angle between the plane AB and CD. Two intersecting planes are said to be perpendicular to each other when the dihedral angle between them is a right angle.
16. Projection:
The projection of a point on a given line or a plane is the foot of the perpendicular drawn from the point to the line or plane. The projection of a line straight or curved on a plane is the locus of the feet of the perpendiculars drawn from all points in the given line to the given plane. It is also called orthogonal projection. In the figure, projections of a curved line AB on plane XY and a line CD are shown as curved line ab and straight line cd.
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