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laystyle p_ 0 x_ 0 y_ 0 z_ 0 and let n a b c displaystyle mathbf n a b c be a nonzero vector the plane determined by this point and vector consists of those points p displaystyle p with position vector r displaystyle mathbf r such that the vector drawn from p 0 displaystyle p_ 0 to p displaystyle p is perpendicular to n displaystyle mathbf n recalling that two vectors are perpendicular if and only if their dot product is zero it follows that the desired plane can be described as the set of all points r displaystyle mathbf r such that n r r 0 0 displaystyle mathbf n cdot mathbf r mathbf r _ 0 0 the dot here means a dot product not scalar multiplication expanded this becomes a x x 0 b y y 0 c z z 0 0 displaystyle a x x_ 0 b y y_ 0 c z z_ 0 0 which is the point normal form of the equation of a plane citation needed this is just a linear equation a x b y c z d 0 where d a x 0 b y 0 c z 0 displaystyle ax by cz d 0 text where d ax_ 0 by_ 0 cz_ 0 conversely it is easily shown that if a b c and d are constants and a b and c are not all zero then the graph of the equation a x b y c z d 0 displaystyle ax by cz d 0 is a plane having the vector n a b c displaystyle mathbf n a b c as a normal citation needed this familiar equation for a plane is called the general form of the equation of the plane 19 in three dimensions lines can not be described by a single linear equation so they are frequently described by parametric equations x x 0 a t displaystyle x x_ 0 at y y 0 b t displaystyle y y_ 0 bt z z 0 c t displaystyle z z_ 0 ct where x y and z are all functions of the independent variable t which ranges over the real numbers x 0 y 0 z 0 is any point on the line a b and c are related to the slope of the line such that the vector a b c is parallel to the line conic sections edit main article conic section in the cartesian coordinate system the graph of a quadratic equation in two variables is always a conic section though it may be degenerate and all conic sections arise in this way the equation will be of the form a x 2 b x y c y 2 d x e y f 0 with a b c not all zero displaystyle ax 2 bxy cy 2 dx ey f 0 text with a b c text not all zero as scaling all six constants yields the same locus of zeros one can consider conics as points in the five dimensional projective space p 5 displaystyle mathbf p 5 the conic sections described by this equation can be classified using the discriminant 20 b 2 4 a c displaystyle b 2 4ac if the conic is non degenerate then if b 2 4 a c 0 displaystyle b 2 4ac 0 the equation represents an ellipse if a c displaystyle a c and b 0 displaystyle b 0 the equation represents a circle which is a special case of an ellipse if b 2 4 a c 0 displaystyle b 2 4ac 0 the equation represents a parabola if b 2 4 a c 0 displaystyle b 2 4ac 0 the equation represents a hyperbola if we also have a c 0 displaystyle a c 0 the equation represents a rectangular hyperbola quadric surfaces edit main article quadric surface a quadric or quadric surface is a 2 dimensional surface in 3 dimensional space defined as the locus of zeros of a quadratic polynomial in coordinates x 1 x 2 x 3 the general quadric is defined by the algebraic equation 21 i j 1 3 x i q i j x j i 1 3 p i x i r 0 displaystyle sum _ i j 1 3 x_ i q_ ij x_ j sum _ i 1 3 p_ i x_ i r 0 quadric surfaces include ellipsoids including the sphere paraboloids hyperboloids cylinders cones and planes distance and angle edit main articles distance and angle the distance formula on the plane follows from the pythagorean theorem in analytic geometry geometric notions such as distance and angle measure are defined using formulas these definitions are designed to be consistent with the underlying euclidean geometry for example using cartesian coordinates on the plane the distance between two points x 1 y 1 and x 2 y 2 is defined by the formula d x 2 x 1 2 y 2 y 1 2 displaystyle d sqrt x_ 2 x_ 1 2 y_ 2 y_ 1 2 which can be viewed as a version of the pythagorean theorem similarly the angle that a line makes with the horizontal can be defined by the formula θ arctan m displaystyle theta arctan m where m is the slope of the line in three dimensions distance is given by the generalization of the pythagorean theorem d x 2 x 1 2 y 2 y 1 2 z 2 z 1 2 displaystyle d sqrt x_ 2 x_ 1 2 y_ 2 y_ 1 2 z_ 2 z_ 1 2 while the angle between two vectors is given by the dot product the dot product of two euclidean vectors a and b is defined by 22 a b d e f a b cos θ displaystyle mathbf a cdot mathbf b stackrel mathrm def left mathbf a right left mathbf b right cos theta where θ is the angle between a and b transformations edit a y f x x b y f x 3 c y f x 3 d y 1 2 f x transformations are applied to a parent function to turn it into a new function with similar characteristics the graph of r x y displaystyle r x y is changed by standard transformations as follows changing x displaystyle x to x h displaystyle x h moves the graph to the right h displaystyle h units changing y displaystyle y to y k displaystyle y k moves the graph up k displaystyle k units changing x displaystyle x to x b displaystyle x b stretches the graph horizontally by a factor of b displaystyle b think of the x displaystyle x as being dilated changing y displaystyle y to y a displaystyle y a stretches the graph vertically changing x displaystyle x to x cos a y sin a displaystyle x cos a y sin a and changing y displaystyle y to x sin a y cos a displaystyle x sin a y cos a rotates the graph by an angle a displaystyle a there are other standard transformation not typically studied in elementary analytic geometry because the transformations change the shape of objects in ways not usually considered skewing is an example of a transformation not usually considered for more information consult the wikipedia article on affine transformations for example the parent function y 1 x displaystyle y 1 x has a horizontal and a vertical asymptote and occupies the first and third quadrant and all of its transformed forms have one horizontal and vertical asymptote and occupies either the 1st and 3rd or 2nd and 4th quadrant in general if y f x displaystyle y f x then it can be transformed into y a f b x k h displaystyle y af b x k h in the new transformed function a displaystyle a is the factor that vertically stretches the function if it is greater than 1 or vertically compresses the function if it is less than 1 and for negative a displaystyle a values the function is reflected in the x displaystyle x axis the b displaystyle b value compresses the graph of the function horizontally if greater than 1 and stretches the function horizontally if less than 1 and like a displaystyle a reflects the function in the y displaystyle y axis when it is negative the k displaystyle k and h displaystyle h values introduce translations h displaystyle h vertical and k displaystyle k horizontal positive h displaystyle h and k displaystyle k values mean the function is translated to the positive end of its axis and negative meaning translation towards the negative end transformations can be applied to any geometric equation whether or not the equation represents a function transformations can be considered as individual transactions or in combinations suppose that r x y displaystyle r x y is a relation in the x y displaystyle xy plane for example x 2 y 2 1 0 displaystyle x 2 y 2 1 0 is the relation that describes the unit circle finding intersections of geometric objects edit main article intersection geometry for two geometric objects p and q represented by the relations p x y displaystyle p x y and q x y displaystyle q x y the intersection is the collection of all points x y displaystyle x y which are in both relations 23 for example p displaystyle p might be the circle with radius 1 and center 0 0 displaystyle 0 0 p x y x 2 y 2 1 displaystyle p x y x 2 y 2 1 and q displaystyle q might be the circle with radius 1 and center 1 0 q x y x 1 2 y 2 1 displaystyle 1 0 q x y x 1 2 y 2 1 the intersection of these two circles is the collection of points which make both equations true does the point 0 0 displaystyle 0 0 make both equations true using 0 0 displaystyle 0 0 for x y displaystyle x y the equation for q displaystyle q becomes 0 1 2 0 2 1 displaystyle 0 1 2 0 2 1 or 1 2 1 displaystyle 1 2 1 which is true so 0 0 displaystyle 0 0 is in the relation q displaystyle q on the other hand still using 0 0 displaystyle 0 0 for x y displaystyle x y the equation for p displaystyle p becomes 0 2 0 2 1 displaystyle 0 2 0 2 1 or 0 1 displaystyle 0 1 which is false 0 0 displaystyle 0 0 is not in p displaystyle p so it is not in the intersection the intersection of p displaystyle p and q displaystyle q can be found by solving the simultaneous equations x 2 y 2 1 displaystyle x 2 y 2 1 x 1 2 y 2 1 displaystyle x 1 2 y 2 1 traditional methods for finding intersections include substitution and elimination substitution solve the first equation for y displaystyle y in terms of x displaystyle x and then substitute the expression for y displaystyle y into the second equation x 2 y 2 1 displaystyle x 2 y 2 1 y 2 1 x 2 displaystyle y 2 1 x 2 we then substitute this value for y 2 displaystyle y 2 into the other equation and proceed to solve for x displaystyle x x 1 2 1 x 2 1 displaystyle x 1 2 1 x 2 1 x 2 2 x 1 1 x 2 1 displaystyle x 2 2x 1 1 x 2 1 2 x 1 displaystyle 2x 1 x 1 2 displaystyle x 1 2 next we place this value of x displaystyle x in either of the original equations and solve for y displaystyle y 1 2 2 y 2 1 displaystyle 1 2 2 y 2 1 y 2 3 4 displaystyle y 2 3 4 y 3 2 displaystyle y frac pm sqrt 3 2 so our intersection has two points 1 2 3 2 and 1 2 3 2 displaystyle left 1 2 frac sqrt 3 2 right text and left 1 2 frac sqrt 3 2 right elimination add or subtract a multiple of one equation to the other equation so that one of the variables is eliminated for our current example if we subtract the first equation from the second we get x 1 2 x 2 0 displaystyle x 1 2 x 2 0 the y 2 displaystyle y 2 in the first equation is subtracted from the y 2 displaystyle y 2 in the second equation leaving no y displaystyle y term the variable y displaystyle y has been eliminated we then solve the remaining equation for x displaystyle x in the same way as in the substitution method x 2 2 x 1 1 x 2 1 displaystyle x 2 2x 1 1 x 2 1 2 x 1 displaystyle 2x 1 x 1 2 displaystyle x 1 2 we then place this value of x displaystyle x in either of the original equations and solve for y displaystyle y 1 2 2 y 2 1 displaystyle 1 2 2 y 2 1 y 2 3 4 displaystyle y 2 3 4 y 3 2 displaystyle y frac pm sqrt 3 2 so our intersection has two points 1 2 3 2 and 1 2 3 2 displaystyle left 1 2 frac sqrt 3 2 right text and left 1 2 frac sqrt 3 2 right for conic sections as many as 4 points might be in the intersection finding intercepts edit main articles x intercept and y intercept one type of intersection which is widely studied is the intersection of a geometric object with the x displaystyle x and y displaystyle y coordinate axes the intersection of a geometric object and the y displaystyle y axis is called the y displaystyle y intercept of the object the intersection of a geometric object and the x displaystyle x axis is called the x displaystyle x intercept of the object for the line y m x b displaystyle y mx b the parameter b displaystyle b specifies the point where the line crosses the y displaystyle y axis depending on the context either b displaystyle b or the point 0 b displaystyle 0 b is called the y displaystyle y intercept tangents and normals edit tangent lines and planes edit main article tangent in geometry the tangent line or simply tangent to a plane curve at a given point is the straight line that just touches the curve at that point informally it is a line through a pair of infinitely close points on the curve more precisely a straight line is said to be a tangent of a curve y f x at a point x c on the curve if the line passes through the point c f c on the curve and has slope f c where f is the derivative of f a similar definition applies to space curves and curves in n dimensional euclidean space as it passes through the point where the tangent line and the curve meet called the point of tangency the tangent line is going in the same direction as the curve and is thus the best straight line approximation to the curve at that point similarly the tangent plane to a surface at a given point is the plane that just touches the surface at that point the concept of a tangent is one of the most fundamental notions in differential geometry and has been extensively generalized see tangent space normal line and vector edit main article normal geometry in geometry a normal is an object such as a line or vector that is perpendicular to a given object for example in the two dimensional case the normal line to a curve at a given point is the line perpendicular to the tangent line to the curve at the point in the three dimensional case a surface normal or simply normal to a surface at a point p is a vector that is perpendicular to the tangent plane to that surface at p the word normal is also used as an adjective a line normal to a plane the normal component of a force the normal vector etc the concept of normality generalizes to orthogonality see also edit cross product rotation of axes translation of axes vector space notes edit boyer carl b 1991 the age of plato and aristotle a history of mathematics second ed john wiley sons inc pp 94 95 isbn 0 471 54397 7 menaechmus apparently derived these properties of the conic sections and others as well since this material has a strong resemblance to the use of coordinates as illustrated above it has sometimes been maintained that menaechmus had analytic geometry such a judgment is warranted only in part for certainly menaechmus was unaware that any equation in two unknown quantities determines a curve in fact the general concept of an equation in unknown quantities was alien to greek thought it was shortcomings in algebraic notations that more than anything else operated against the greek achievement of a full fledged coordinate geometry boyer carl b 1991 apollonius of perga a history of mathematics second ed john wiley sons inc pp 142 isbn 0 471 54397 7 the apollonian treatise on determinate section dealt with what might be called an analytic geometry of one dimension it considered the following general problem using the typical greek algebraic analysis in geometric form given four points a b c d on a straight line determine a fifth point p on it such that the rectangle on ap and cp is in a given ratio to the rectangle on bp and dp here too the problem reduces easily to the solution of a quadratic and as in other cases apollonius treated the question exhaustively including the limits of possibility and the number of solutions boyer carl b 1991 apollonius of perga a history of mathematics second ed john wiley sons inc pp 156 isbn 0 471 54397 7 the method of apollonius in the conics in ma...