2nd topic
2.1. Simple shapes,
their projections and parameters
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Picture 2.1. Polyhedron models and their wireframes.
2.1.1. Polyhedrons
2.1.1. Polyhedrons
The term polyhedron is formed from the roots "poly" or "many" + "hedra" or "side". The image of each figure is a projection of its external surface, which consists of sides. Edges and vertices form the mesh of an outer surface. Projections of this mesh are used in drawings (Pic. 2.1).
Picture 2.2. Prism
2.1.1.1. A prism is a polyhedron with two identical polygons as bases: quadrangles, rectangles or parallelograms. Prisms can be characterized by the form of the base. We have triangular and four-, five-, etc. angled prisms.
The vertical edges of a prism form rectangular faces, which are perpendicular to the bases (Pic. 2.2). Different types of prism (Pic. 2.3, a), their shaded bodies (Pic. 2.3, b), and their wireframes (Pic. 2.3, c) are shown below.
Picture 2.3: a)
The sides of an inclined prism are parallelograms which aren't perpendicular to the bases.
b) orthogonal projections of solid prisms
c) prism wireframes
Picture 2.4. Pyramid.
2.1.1.2. A pyramid is a polyhedron whose sides are triangles that converge at a point or apex. The base of a pyramid can be triangular, square, pentagonal, or any other polygonal shape. A pyramid has at least three sides (at least four faces, including the base). A square pyramid (with a square base and four triangular faces) is one of the most common types (Pic. 2.4).
2.1.1.3. The names of the elements of polyhedra are shown in the picture below. Drag the name of each element to its correct place and check your results.
2.1.2. Shapes of rotation
2.1.2. Shapes of rotation
Shapes of rotation are shapes created by rotating a 2D figure around an axis located on the figure to form the side of the new shape.
Picture 2.5. Cylinder
2.1.2.1. A cylinder (Pic. 2.5) is one of the most basic curvilinear geometric shapes. It is the idealized version of a solid physical tin can having lids on top and bottom. More advanced mathematics now focuses on an infinite curvilinear surface, which is how a cylinder is defined in modern branches of geometry and topology.
Picture 2.6. Cone
2.1.2.2. A cone (Pic. 2.6) is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex.
In geometry, a (general) conical surface is the unbounded surface formed by the union of all the straight lines that pass through a fixed point (the apex or vertex) and any point of some fixed space curve (the directrix) that does not contain the apex. Each of those lines is called a generatrix of the surface.
Picture 2.7. Sphere
2.1.2.3. A sphere (Pic. 2.7) is an object in three-dimensional space representing the surface of a ball.
Like a circle in a two-dimensional space, a sphere is mathematically defined as the set of points at the same distance r from a given point in three-dimensional space. This distance r is the radius of the ball, which is made up of all the points with a distance less than (or, for a closed ball, less than or equal to) r from the given point, the centre of the mathematical ball. These are also referred to as the radius and centre of the sphere, respectively. The most extended straight line segment through the ball, connecting two points of the sphere, passes through the centre, and its length, the diameter of both the sphere and its ball, is thus twice the radius.
Picture 2.8. Torus
2.1.2.4. In geometry, a torus (Pic. 2.8) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis coplanar with the circle. If the axis of revolution does not touch the circle, the surface has a ring shape called a torus of revolution.
2.1.2.5. As the distance to the axis of revolution decreases, the ring torus becomes a horn torus, then a spindle torus, and finally, a sphere (Pic. 2.9, a). See the formation of spherical surfaces (Pic. 2.9, b) and compare the images with different shapes of rotation and rotation axis locations (Pic. 2.9, c).
Picture 2.9: a) spherical surfaces,
b) formation of spherical surfaces,https://en.wikipedia.org/wiki/Torus
c) wireframe models of spherical surfaces.
2.1.2.6. Match the name of the 2D shape with the 3D shape formed by rotating it.
2.1.3. Shape parameters
2.1.3. Shape parameters
2.1.3.1. Shape parameters are dimensions in the appropriate units (millimetre - mm, centimetre - cm, meter - m) used to determine the shape's size and orientation and the location of an element or part (Pic. 2.10):
a - length
b - width
h - height
Picture 2.10. Parameters for a prism and pyramid
2.1.3.2. The orthogonal projections of regular shapes have an axis of symmetry, and rotated shapes have an axis of rotation.
2.1.3.3. Standard symbols can also be added to a dimension to supplement and explain it (see Pic. 2.11):
A square symbol indicates that the length and height are the same.
A circle symbol indicates a diameter.
Picture 2.11. Orthogonal projections of shapes with parameters and symbols
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