数学知识

三角函数:
在一个直角三角形中，设∠C=90°，∠A, B, C 所对的边分别记作 a,b,c，那么对于锐角∠A，它的对边 a 和斜边 c 的比值 a/c 叫做∠A的正弦，记作 sinA；它的邻直角边 b 和斜边 c 的比值 b/c 叫做∠A的余弦，记作 cosA；它的对边 a 和邻直角边 b 的比值 a/b 叫做∠A的正切，记作 tanA

θ
θ 希腊字母

Θ 西塔

Theta（大写Θ,小写θ）,在希腊语中,是第八个希腊字母.
大写的Θ是：

粒子物理学中pentaquark用Θ+来表示

小写的θ是：

数学上常代表平面的角

万花尺方程

数学原理视频讲解

https://www.bilibili.com/video/BV1w7411q7xH

利用turtle画圆

半径为r的圆的方程
x = r* cos(θ)
y = r* sin(θ)

# Draw the circle by turtle
import math
import turtledef drawCircleTurtle(x,y,r):# Move the turtle to the jumping-off pointturtle.up()turtle.setpos(x+r,y)turtle.down()# Draw the circlefor i in range(0,365,5):  # 从0到360,每隔5步循环 i是角度参数a = math.radians(i)   # 将角度转化为弧度#print(a)# python中sin和cos都是以弧度为参数,所以需要转化为弧度.turtle.setpos(x+r*math.cos(a), y+r*math.sin(a))print(x+r*math.cos(a))drawCircleTurtle(100,100,50)
turtle.mainloop()

输出结果:

创建螺线

turtle 模块用于绘图
pillow python图像库(PIL)的一个分支,用于保存螺线图像

Spiro 构造函数

#万花尺
import sys , random,argparse
import turtle
import numpy as np
import math
from PIL import Image
from datetime import datetime
from fractions import gcd# a class for draw a spirograph
class Spiro:def __init__(self,xc,yc,col,R,r,l):self.t = turtle.Turtle()self.t.shape('turtle')self.step = 5self.drawingComplete = Falseself.setparams(xc,yc,col,R,r,l)self.restart()# set the parametersdef setparams(self,xc,yc,col,R,r,l):# the Spirograph paramentersself.xc = xcself.yc = ycself.R = int(R)self.r = int(r)self.l = lself.col = col# reduce r/R to its smallest form by dividing with the GCDgcdVal = math.gcd(self.r, self.R)    #gcd 最大公约数 greatest common divisorself.nRot = self.r//gcdVal  # //取整数# get ratio of radii (获得半径曲率)self.k = r/float(R)#set the colorself.t.color(*col)#store the current angleself.a = 0#restart the drawingdef restart(self):#set the flagself.drawingComplete = False#show the turtleself.t.showturtle()#go to the first pointself.t.up()R, k, l = self.R, self.k, self.la = 0.0x = R*((1-k)*math.cos(a) + l*k*math.cos((1-k)*a/k))y = R*((1-k)*math.sin(a) + l*k*math.sin((1-k)*a/k))self.t.setpos(self.xc +x, self.yc + y)self.t.down()#draw the whole thingdef draw(self):#draw the rest of the pointsR, k, l = self.R, self.k, self.lfor i in range(0, 360*self.nRot + 1, self.step):a = math.radians(i)x = R * ((1 - k) * math.cos(a) + l * k * math.cos((1 - k) * a / k))y = R * ((1 - k) * math.sin(a) + l * k * math.sin((1 - k) * a / k))self.t.setpos(self.xc + x, self.yc + y)#drawing is now done so hide the turtle cursorself.t.hideturtle()#update by one setupdef update(self):#skip the rest of the steps if doneif self.drawingComplete:return#increment the angleself.a += self.step#draw a stepR, k, l = self.R, self.k, self.l#set the anglea = math.radians(self.a)x = self.R * ((1 - k) * math.cos(a) + l * k * math.cos((1 - k) * a / k))y = self.R * ((1 - k) * math.sin(a) + l * k * math.sin((1 - k) * a / k))self.t.setpos(self.xc + x, self.yc +y)#if drawing is complete, set the flagif self.a >= 360*self.nRot:self.drawingComplete = True#drawing is now done so hide the turtle cursorself.t.hideturtle()def clear(self):self.t.clear()# a class fro animating Spirographs
class SpiroAnimator:#constructordef __init__(self, N):#set the timer value in millisecondsself.deltaT = 10#get the window dimensionsself.width = turtle.window_width()self.height = turtle.window_height()#Create the Spiro objectsself.spiros = []for i in range(N):#generate random parametersrparams = self.genRandomParams()#set the Spiro Parametersspiro = Spiro(*rparams)self.spiros.append(spiro)#call timerturtle.ontimer(self.update, self.deltaT)#restart spiro drawingdef restart(self):for spiro in self.spiros:#clearspiro.clear()#generate random parametersrparams = self.genRandomParams()#set the spiro paramentersspiro.setparams(*rparams)#restart drawingspiro.restart()
# generate random parametersdef genRandomParams(self):width, height = self.width, self.heightR = random.randint(50, min(width, height) // 2)r = random.randint(10, 9 * R // 10)l = random.uniform(0.1, 0.9)xc = random.randint(-width // 2, width // 2)yc = random.randint(-height // 2, height // 2)col = (random.random(),random.random(),random.random())return (xc, yc, col, R, r, l)def update(self):#update all spirosnComplete = 0for spiro in self.spiros:#updatespiro.update()#count completed spirosif spiro.drawingComplete:nComplete += 1#restart if all spiros are completeif nComplete == len(self.spiros):self.restart()#call the timerturtle.ontimer(self.update, self.deltaT)#toggle turtle cursor on and offdef toggleTurtles(self):for spiro in self.spiros:if spiro.t.isvisible():spiro.t.hideturtle()else:spiro.t.shwoturtle()# save drawing as PNG file
def saveDrawing():#hide the turtle cursorturtle.hideturtle()#generate unique filenamesdateStr = (datetime.now()).strftime("%d%b%Y-%H%M%S")fileName = 'Spiro-' + dateStrprint('saving drawing to %s.eps/png' % fileName)#get the tkinter canvascanvas = turtle.getcanvas()#save the drawing as a postscript imagecanvas.postscript(file = fileName +'.eps')#use the Pillow module to convert the postscript image file to PNGimg = Image.open(fileName + '.eps')img.save(fileName + '.png', 'png')#show the turtle cursorturtle.showturtle()
#main() function
def main():# use sys.argv if neededprint('generating spirograph...')#create parserdescStr = '''This program drawsSpirographs using the Turtle module.When run with no arguments, this program draws random Spirographs.Terminology:R:radius of outer circler: radius of inner circlel: ratio of hole distance to r'''parser = argparse.ArgumentParser(description=descStr)parser.add_argument('--sparams',nargs=3,dest='sparams',required=False,help='The three arguments in sparams: R,r,l.')#parse argsargs= parser.parse_args()#set the width of the drawing window to 80 percent of the screen widthturtle.setup(width=0.8)#set the cursor shape to turtleturtle.shape('turtle')#set the title to Spirographs!turtle.title('Spirographs!')#add the key handler to  save our drawingsturtle.onkey(saveDrawing,'s')#start listeningturtle.listen()#hide the main turtle cursorturtle.hideturtle()#check for any arguments sent to --sparams and draw the Spirographif args.sparams:params = [float(x) for x in args.sparams]#draw the Spirograph with the given paramenterscol = (0.0,0.0,0.0)spiro = Spiro(0,0,col,*params)spiro.draw()else:#create the animator objectspiroAnim = SpiroAnimator(4)#add a key handler to toggle the turtle cursorturtle.onkey(spiroAnim.toggleTurtles,'t')#add a key handler to restart to restart the animationturtle.onkey(spiroAnim.restart,'space')# start the turtle main loopturtle.mainloop()
#call main
if __name__=='__main__':main()

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