Analytical Geometry | ClipArt ETC

This mathematics ClipArt gallery offers 87 illustrations of analytical geometry, which is also called coordinate geometry, Cartesian geometry, algebraic geometry, or simply analytic geometry. It is the study of geometry using principles of algebra.

Finding the angle between two lines whose equations are in intercept form.

Angle

Finding the angle between two lines whose equations are in intercept form.

Transforming from one set of rectangular axes to another with the same origin, but different direction.

Rectangular Axes

Transforming from one set of rectangular axes to another with the same origin, but different direction.

Finding the distance between two points using polar coordinates.

Distance Between

Finding the distance between two points using polar coordinates.

Multiple lines showing many bisectors on the coordinate plane.

Line Bisectors

Multiple lines showing many bisectors on the coordinate plane.

Illustration showing the periodic center of motion as it often happens when two positions of a line are known and they are moving in the same plane and we wish to find an axis about which this line could revolve to occupy the two given positions.

Periodic Center of Motion

Illustration showing the periodic center of motion as it often happens when two positions of a line…

Aronhold stated if any three bodies have plane motion their three virtual centers are three points on one straight line

Centrodes

Aronhold stated if any three bodies have plane motion their three virtual centers are three points on…

Illustration showing a centroid, "the curve passing through the successive positions of the instantaneous centre of a body having a combined motion of rotation and translation is called a centroid." A smooth curve passes through the successive positions of the instantaneous centers (all the centers marked o)will be the centroid ab.

Centroid

Illustration showing a centroid, "the curve passing through the successive positions of the instantaneous…

The chord of a circle.

Circle Chord

The chord of a circle.

A circle and triangle situated on coordinate planes.

Circle

A circle and triangle situated on coordinate planes.

Intersection of lines between a circle and its polar point.

Circle Polar Point

Intersection of lines between a circle and its polar point.

A circle in its common, or central form. This is used to assist students in finding the equation of any circle.

Common Form of Circle

A circle in its common, or central form. This is used to assist students in finding the equation of…

Illustration showing complex numbers with a modulus equal to unity. The lines representing these numbers terminate in points lying on the circumference of a circle whose radius is unity.

Geometric Inspection of Complex Numbers

Illustration showing complex numbers with a modulus equal to unity. The lines representing these numbers…

Illustration showing a conchoid, "a curve, shell-like in flexure (whence the name), invented by Nicomedes in the 2nd century B.C., and used by him for finding two mean proportionals."

Conchoid

Illustration showing a conchoid, "a curve, shell-like in flexure (whence the name), invented by Nicomedes…

Conic Motion

Instantaneous axis of two cones, each with angular velocity

Finding the equation of a line with polar coordinates.

Polar Coordinates

Finding the equation of a line with polar coordinates.

Changing from polar to rectangular coordinates.

Rectangular Coordinates

Changing from polar to rectangular coordinates.

Illustration showing conchoidal curves.

Conchoidal Curves

Illustration showing conchoidal curves.

Illustration showing conchoidal curves.

Conchoidal Curves

Illustration showing conchoidal curves.

Illustration showing confocal curves.

Confocal Curves

Illustration showing confocal curves.

Illustration showing equilateral curves.

Equilateral Curves

Illustration showing equilateral curves.

"Since these curves are not closed, one pair cannot be used for continuous motion; but a pair of such curves may be well adapted to sectional wheels requiring a varying angular velocity." This figure shows an example of the process.

Motion of Open Curves

"Since these curves are not closed, one pair cannot be used for continuous motion; but a pair of such…

Illustration showing Pascal's Volute curves.

Pascal's Volute Curves

Illustration showing Pascal's Volute curves.

Illustration showing trajectory curves.

Trajectory Curves

Illustration showing trajectory curves.

Illustration showing a cycloid curve. "The curve generated by a point in the plane of a circle when the circle is rolled along a straight line and always in the same plane."

Cycloid

Illustration showing a cycloid curve. "The curve generated by a point in the plane of a circle when…

Illustration showing a cycloid curve. "The curve generated by a point in the plane of a circle when the circle is rolled along a straight line and always in the same plane."

Cycloid

Illustration showing a cycloid curve. "The curve generated by a point in the plane of a circle when…

An illustration showing how to construct a cycloid. "The circumference C=3.14D. Divide the rolling circle and base line C into a number of equal parts, draw through the division point the ordinates and abscissas, make aa' = 1d, bb' = 2'e, cc = 3f, then ab' and c' are points in the cycloid. In the Epicycloid and Hypocycloid the abscissas are circles and the ordinates are radii to one common center."

Construction Of A Cycloid

An illustration showing how to construct a cycloid. "The circumference C=3.14D. Divide the rolling circle…

Illustration showing cycloid curves. "The curve generated by a point in the plane of a circle when the circle is rolled along a straight line and always in the same plane."

Cycloids

Illustration showing cycloid curves. "The curve generated by a point in the plane of a circle when the…

Showing distance between two points on a coordinate plane.

Distance

Showing distance between two points on a coordinate plane.

Finding the perpendicular distance of points whose equation is x cosa + y sina = p.

Perpendicular Distance

Finding the perpendicular distance of points whose equation is x cosa + y sina = p.

Each section is an ellipse. The surface is generated by an ellipse moving parallel to itself along two ellipses as directices.

Ellipsoid

Each section is an ellipse. The surface is generated by an ellipse moving parallel to itself along two…

Tangent lines are drawn from a given external point to a circle. The equation is found by joining their points of contact

Chord Equation

Tangent lines are drawn from a given external point to a circle. The equation is found by joining their…

Finding the equation of a circle with the origin as its center.

Circle Equation

Finding the equation of a circle with the origin as its center.

To find the equation of a straight line in "tangent" form.

Line Equation

To find the equation of a straight line in "tangent" form.

Finding the equation of a line through two given points.

Line Equation

Finding the equation of a line through two given points.

Finding the equation of the normal at any point on a circle.

Normal Equation

Finding the equation of the normal at any point on a circle.

Finding the polar equation of a straight line.

Polar Equation

Finding the polar equation of a straight line.

Other forms of line equations.

Line Equations

Other forms of line equations.

An illustration showing how to construct an evolute of a circle. "Given the pitch p, the angle v, and radius r. Divide the angle v into a number of equal parts, draw the radii and tangents for each part, divide the pitch p into an equal number of equal parts, then the first tangent will be one part, second two parts, third three parts, etc., and so the Evolute is traced."

Construction Of An Evolute Of A Circle

An illustration showing how to construct an evolute of a circle. "Given the pitch p, the angle v, and…

Polar property example.

Example

Polar property example.

Polar coordinate example. Ex. (4, 60 degrees)

Coordinate Example

Polar coordinate example. Ex. (4, 60 degrees)

Example of an equation with a given external point.

Equation Example

Example of an equation with a given external point.

"Folium of Descartes, with its asymptote. The equation is (4-y)(y-1)<sup>2</sup> = 3x<sup>2</sup>y ... In geometry, a plane cubic curve having a crunode, and one real inflexion, which lies at infinity.

Folium of Descartes

"Folium of Descartes, with its asymptote. The equation is (4-y)(y-1)2 = 3x2y ... In geometry, a plane…

A line with a standard form equation.

General Form

A line with a standard form equation.

A line with intercept form equation.

Intercept Form

A line with intercept form equation.

A line on the coordinate plane showing evidence of "perpendicular" form.

Perpendicular Form

A line on the coordinate plane showing evidence of "perpendicular" form.

Generating a hyperbola from two equal and parallel circular disks.

Generate Hyperbola

Generating a hyperbola from two equal and parallel circular disks.

Three dimensional representation of a variable hyperbola moving parallel to itself along the parabolas as directrices.

Hyperbolic Parabaloid

Three dimensional representation of a variable hyperbola moving parallel to itself along the parabolas…

Two dimensional representation of variable hyperbola moving parallel to itself along the parabolas as directrices.

Hyperbolic Parabaloid

Two dimensional representation of variable hyperbola moving parallel to itself along the parabolas as…

Equation of a line with two given points and a given inclination.

Given Incline

Equation of a line with two given points and a given inclination.

An illustration showing how to use isometric perspective. "This kind of perspective admits of scale measurements the same as any ordinary drawing, and gives a clear representation of the object. It is easily learned. All horizontal rectangular lines are drawn at an angle of 30°. All circles are ellipses of proportion, as shown."

Construction Using Isometric Perspective

An illustration showing how to use isometric perspective. "This kind of perspective admits of scale…

A Lemniscate is, in general, a curve generated by a point moving so that the product of its distances from two fixed points is the square of half the distance between the points. It is a particular case of the Cassinian oval and resembles a figure 8. When the line joining the two fixed points is the axis of x and the middle point of this line is the origin, the Cartesian equation is the fourth degree equation, (((x^2)+(y^2))^2)=2(a^2)((x^2)-(y^2)). The polar equation is (ℽ^2) = 2(a^2)cos(2θ). The locus of the feet of the perpendiculars from the center of an equilateral hyperbola to its tangents is a lemniscate. The name lemniscate is sometimes given to any crunodal symmetric quartic curve having no infinite branch. The name is also sometimes given to a general class of curves derived from other curves in the way that the above is derived from the equilateral hyperbola. With these more general definitions of the lemniscate the above curve is called the lemniscate of Bernoulli.

Lemniscate

A Lemniscate is, in general, a curve generated by a point moving so that the product of its distances…

Finding the length of a tangent from a given point to a circle.

Tangent Length

Finding the length of a tangent from a given point to a circle.

A positive slope line making a 60 degree angle with the x-axis.

Straight Line

A positive slope line making a 60 degree angle with the x-axis.

Finding the polar equation of a straight line by passing through two given points.

Straight Line

Finding the polar equation of a straight line by passing through two given points.

Another method for using perpendicular form.

Alternate Method

Another method for using perpendicular form.

The aggregate of the components of momentum.

Components of Momentum

The aggregate of the components of momentum.

Illustration showing two points a and b to be in the same plane and parallel.

Motions Of 2 Points In Same Plane And Parallel

Illustration showing two points a and b to be in the same plane and parallel.

Illustration showing two points a and b to be in the same plane and parallel.

Motions Of 2 Points In Same Plane And Parallel

Illustration showing two points a and b to be in the same plane and parallel.

Illustration showing three points a, b, and c in motion. The magnitude and direction of a and b are used to find the l.v. of c.

Motions Of 3 Points

Illustration showing three points a, b, and c in motion. The magnitude and direction of a and b are…

Parallelopiped of Motions

Illustration used "If three component motions ab, ac, and ad are combined, their resultant af will be…

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