The vertebral column or spine is a flexible column that supports the head, neck, and body and  allows for their movements. It also protects the spinal cord.
Regions of the Vertebral Column
In an adult, the spine is subdivided into five regions: the cervical, the thoracic, the lumbar, the sacral, and the coccygeal region. The spine initially develops as a series of 33 vertebrae; after 20 years of age, the nine bones in the sacral region, five sacral, and four coccygeal bones fuse to form the...

A hyperbola is a conic section produced when a double-napped cone is intersected by a plane at an angle steeper than the slope of the cone, such that it cuts through both nappes. This intersection yields two separate, mirror-image curves known as branches, which open away from each other along the transverse axis. The nearest points on each branch to the hyperbola’s center are termed vertices, and the distance from the center to a vertex is denoted by a. Perpendicular to the transverse axis is...

A hyperbola consists of all points where the absolute difference of distances to two fixed points, called foci, remains constant. The standard equation isEach branch extends infinitely and approaches two asymptotes, which guide the curve’s behavior. The parameters a and b define key features: a measures the distance from the center to each vertex along the transverse axis, while b influences the slopes of the asymptotes. The asymptotes have equationsA rectangle centered at the origin with...

The superior view of the cranium shows the frontal and paired parietal bones.
The frontal bone is the single bone that forms the forehead. At its anterior midline, between the eyebrows, there is a slight depression called the glabella. The frontal bone also forms the supraorbital margin of the orbit. Near the middle of this margin is the supraorbital foramen, the opening that provides passage for a sensory nerve to the forehead. The frontal bone is thickened just above each supraorbital margin,...

A typical vertebra, with the exception of the sacrum and coccyx, consists of a body, a vertebral arch, and seven different projections termed processes. The anterior portion of the vertebrae, the body, supports about half the body’s weight. The vertebral bodies progressively increase in size and thickness from the cervical region to the lumbar region of the vertebral column. The intervertebral discs present between the bodies of adjacent vertebrae firmly unites them, forming a continuous column.

In addition to being held together by the intervertebral discs, adjacent vertebrae also articulate with each other at synovial joints formed between the superior and inferior articular processes called zygapophysial joints (facet joints). These are plane joints that provide for only limited motions between the vertebrae. The orientation of the articular processes at these joints varies in different regions of the vertebral column and serves to determine the types of motions available in each...