Polar coordinate solution, how to solve this problem with polar coordinate method

The two ends of a line segment with a length of 2a slide on the two straight lines that intersect vertically at o, and the vertical line from o to the line segment is used to find the polar coordinate equation of the trajectory of the perpendicular foot.

Solution: In order to simplify the problem, take two perpendicular straight lines as the coordinate axes, the line segment ab = 2a, point A slides on the x-axis, and point B is at y

Slide on the axis. Take the O point as the pole and the Ox axis as the polar axis; From O to AB as the perpendicular line, C is the perpendicular foot; then oc= , aoc= ; Establish.

The Cartesian coordinates of point a are (m,0); The Cartesian coordinates of point b are (0,n), oba= aoc= ; Therefore m=2asin and n=2acos

Therefore, the polar coordinate equation for perpendicular foot c is: =mcos =2asin cos =asin2 ; i.e. =asin2 is the polar equation for perpendicular foot c.

of which 0 < 90

We can set the vertical foot as p(rcos, rsin) and o as the origin, and use two straight lines to establish a Cartesian coordinate system if the line segment is parallel to one of the two straight lines, then there must be an endpoint at the origin, and the vertical foot is the origin o(0,0).

If the line segment is not parallel to either of the two lines, and the vector op is perpendicular to the line segment, then op should be parallel to the line segment normal, and the line passes through point p, then the line L equation in which the line segment is located is.

rcos (x-rcos)+rsin(y-rsin)=0y=0, x=r cos

When x=0, y=r sin

The intersection point of the line and the coordinate axis is m(r cos,0) and n(0,r sin) and the length of the line segment is 2a, then [r cos -0) 2+(0-r sin) 2]=2a, and the arrangement obtains, r=asin2 ,0 x<2 )

Polar coordinates are made x=rcos@, y=rsin@, and then brought into the expression of the original Cartesian coordinate system.

So for this problem, bring in (x-1) +y-1) =2. You can first remove the brackets and sort it out, that is, x +y =2 (x + y), so that because x = rcos@, y = rsin@, so x +y = r, and then it becomes r = 2r (cos@ + ysin@), and remove an r on both sides at the same time to get the final result r=2 (cos@ + ysin@).

When converting to polar fiber coordinates, you have to draw a straight line from the origin of the coordinates pointing to the square direction of the x-axis, and then rotate it counterclockwise in the integral area to the negative direction of x. The range of angles depends on the angle of rotation, and the maximum range is [0,pi] (from the positive direction of the x-axis to the negative direction of the x-axis).

Converted into Cartesian coordinate equations, the former is a circle x2+y2=8, and the latter is a straight line x-y-2=0The distance from the center of the circle to the straight line is the root number 2, and all have 3 points.

The inclination angle of the straight line is 3, and in the polar coordinate system, the constant represents the ray, so the polar coordinate system of the line is 3 and 4 3, or it is written as tan 3, or ( 3cos sin ) 0

When you know the angle or the distance to the pole, it is better to use the polar equation, the circle equation on the polar axis is p=2rcos (r is the radius of the circle, substitution is sufficient) and the circular equation perpendicular to the polar axis is p=2rsin (r is the radius of the circle, substitution is sufficient).

If not on the polar axis or perpendicular to the polar axis.

Then the general equation for a circle is: p -p -2pp cos( -=r (p is the distance from a known point to the pole, is the angle between the known point and the pole, and r is the radius of the circle).

Conic curve: The unified polar coordinate equation for (quadratic non-circular curve) is =ep (1-e cos) where e is the eccentricity and p is the distance from the focal point to the alignment.

In 3D graphics, it is best to use polar coordinates to solve problems. There's it in the book. Dear, read more books, look at the study guide books for problem solving.

You can do the math and you can change it.

If you assume that the x-axis is the axis of the polar coordinates, then the position of each point is represented by the polar coordinates and you will have two parameters, and it is OK to solve the right triangle and find its side length.

The ratio of diameter to circumference is used for calculation, so the physical aspect is more inclined to use radians.

Converts between polar and planar Cartesian coordinate systems.

The two coordinates r and in the polar coordinate system can be converted to coordinate values in the Cartesian coordinate system by the following formula.

x = r*cos( )y = r*sin( ) From the above two formulas, we can get how to calculate the coordinates under polar coordinates from the x and y coordinates in the Cartesian coordinate system.

r = sqrt(x^2 + y^2),= arctan y/x

In the case of x = 0: if y is positive = 90° ( 2 radians); If y is negative, then = 270° (3 2 radians)

You can set the formula x=r cos

y=r sinθ

r^2=x^2+y^2

tan = y x [Cartesian coordinates polar coordinates}

If you have any questions, please feel free to ask.

Law One. <>

Law II. <>

According to symmetry.

d (x²＋y²)dσ=8∫∫d' (x²＋y²)dσd':x y 1,0 x 2,0 y x into polar coordinates x=rcos, y=rsin, substituting the integration region d'Available.

r²≥1 → r≥1

0≤rcosθ≤2 → 0≤r≤2secθ0≤rsinθ≤rcosθ →0≤θ≤4

So [0, 4],r [1,2sec ]8 d' (x²＋y²)dσ

8∫(0，π/4) dθ∫(1，2secθ) r³dr=2∫(0，π/4) [16(sec²θ)1]dθ=32∫(0，π/4) (tan²θ＋1)d(tanθ)–2 ①=32(1/3 tan³θ＋tanθ)|0, 4) 2Note: (sec ) 4d = sec d(tan ) = tan 1) d(tan ).