Conic sections are the family of curves that result from the intersection of a plane with a cone. Depending on the angle at which the plane intersects the cone, the resulting curve will be an circle, ellipse, parabola or a hyperbola. It turns out that the general definition of a conic section depends on the distance between a fixed point, called the focus, and a fixed line, called the directrix.
An orientation chooses which of the two half-lines determined by O is the positive, and which is negative; we then say that the line "is oriented" or "points" from the negative half towards the positive half. A line with a chosen Cartesian system is called a number line.
Every real number has a unique location on the line. Conversely, every point on the line can be interpreted as a number in an ordered continuum such as the real numbers. The point where the axes meet is taken as the origin for both, thus turning each axis into a number line.
For any point P, a line is drawn through P perpendicular to each axis, and the position where it meets the axis is interpreted as a number. The two numbers, in that chosen order, The cartisian circle the Cartesian coordinates of P.
The reverse construction allows one to determine the point P given its coordinates. The first and second coordinates are called the abscissa and the ordinate of P, respectively; and the point where the axes meet is called the origin of the coordinate The cartisian circle.
The coordinates are usually written as two numbers in parentheses, in that order, separated by a comma, as in 3, Thus the origin has coordinates 0,0and the points on the positive half-axes, one unit away from the origin, have coordinates 1,0 and 0,1. In mathematics, physics, and engineering, the first axis is usually defined or depicted as horizontal and oriented to the right, and the second axis is vertical and oriented upwards.
However, in some computer graphics contexts, the ordinate axis may be oriented downwards. The origin is often labeled O, and the two coordinates are often denoted by the letters X and Y, or x and y.
The axes may then be referred to as the X-axis and Y-axis.
The choices of letters come from the original convention, which is to use the latter part of the alphabet to indicate unknown values. The first part of the alphabet was used to designate known values. A Euclidean plane with a chosen Cartesian coordinate system Cartesian plane.
In a Cartesian plane one can define canonical representatives of certain geometric figures, such as the unit circle with radius equal to the length unit, and center at the originthe unit square whose diagonal has endpoints at 0,0 and 1,1the unit hyperbolaand so on.
The two axes divide the plane into four right anglescalled quadrant. The quadrants may be named or numbered in various ways, but the quadrant where all coordinates are positive is usually called the first quadrant.
If the coordinates of a point are x,ythen its distances from the X-axis and from the Y-axis are y and xrespectively; where The tick marks on the axes are one length unit apart. A Cartesian coordinate system for a three-dimensional space consists of an ordered triplet of lines the axes that go through a common point the originand are pair-wise perpendicular; an orientation for each axis; and a single unit of length for all three axes.
As in the two-dimensional case, each axis becomes a number line. For any point P of space, one considers a plane through P perpendicular to each coordinate axis, and interprets the point where that plane cuts the axis as a number.
The Cartesian coordinates of P are those three numbers, in the chosen order. The reverse construction determines the point P given its three coordinates. Alternatively, each coordinate of a point P can be taken as the distance from P to the plane defined by the other two axes, with the sign determined by the orientation of the corresponding axis.
Each pair of axes defines a coordinate plane. These planes divide space into eight trihedracalled octants. The coordinates are usually written as three numbers or algebraic formulas surrounded by parentheses and separated by commas, as in 3, Thus, the origin has coordinates 0,0,0and the unit points on the three axes are 1,0,00,1,0and 0,0,1.
There are no standard names for the coordinates in the three axes. The coordinates are often denoted by the letters X, Y, and Z or x, y, and zin which case the lines are called the X- Y- and Z-axis, respectively.
In mathematics, physics, and engineering contexts, the first two axes are often defined or depicted as horizontal, with the third axis pointing up. In that case the third coordinate may be called height or altitude. The orientations are usually chosen so that the 90 degree angle from the first axis to the second axis looks counter-clockwise when seen from the point 0,0,1 ; a convention that is commonly called the right hand rule.
The coordinate surfaces of the Cartesian coordinates x, y, z. The z-axis is vertical and the x-axis is highlighted in green. Higher dimensions[ edit ] A Euclidean plane with a chosen Cartesian system is called a Cartesian plane.
Since Cartesian coordinates are unique and non-ambiguous, the points of a Cartesian plane can be identified with pairs of real numbers ; that is with the Cartesian product R.For example, to draw a line you will specify the coordinates of the line’s start point and end point; to draw a circle you will specify the coordinates of the circle’s center and radius.
With DraftSight’s ample tools, you will not need to rely solely on entering coordinates manually, but will be able to use the mouse for much of your drawing. For the cart2sph function, elevation is measured from the x-y plane. Notice that if elevation = 0, the point is in the x-y plane. If elevation = pi/2, then the point is on the positive z-axis.
Extended Capabilities. Tall Arrays Calculate with arrays that have more rows than fit in memory. The calculator will convert the polar coordinates to rectangular (Cartesian) and vice versa, with steps shown. TEACHER ELIGIBILITY TEST SEPTEMBER PAPER II SYLLABUS (Content of School Subjects is as per old syllabus) I.
CHILD DEVELOPMENT AND PEDAGOGY (Marks: 30).
View Notes - Radian Measure and Angles on the Cartesian plane from MHF 4U at Wexford Collegiate School for the Arts. Unit 5: Trigonometric Functions Date: _ L2: Radian Measure and Angles on the. Polar Coordinates, Parametric Equations r Pola tes Coordina distance 2 from the origin, so r = 2 describes the circle of radius 2 with center at the origin.
EXAMPLE Graph the curve given by r = 1 + cosθ. We ﬁrst consider y = 1+ cosx, as in ﬁgure As θ .