Texts and Transformations II

The problem

This example is similar to the one Texts and Transformations, beside that the x-axis and the y-axis have different scalings. To be precise, the ratio of the two scalings is equal to 4. The result is that rotating the text and the image will also stretch these objects.

Here is the code for this example.

(function(){
    var brd, txt, im, p0, p1;
    
    brd = JXG.JSXGraph.initBoard('box1', {boundingbox:[-5,20,5,-20], axis:true, showNavigation:true, showCopyright:true});
    p0 = brd.create('point', [0,0], {style:5, name:'offset'}); 
    p1 = brd.create('point', [3,0], {style:5, name:'rotate'}); 
    txt = brd.create('text',[0,0, 'Hello'], {fontSize:64});
    im = brd.create('image',["/distrib/images/uccellino.jpg", [0,0], [3,12]], {opacity:0.5});

    var tOff = brd.create('transform', [function(){return p0.X()},function(){return p0.Y()}], {type:'translate'}); 
    tOff.bindTo(txt);
    tOff.bindTo(im);
    tOff.bindTo(p1);

    // Rotate text around point "offset" by dragging point "rotate"
    var tRot = brd.create('transform', [function(){return Math.atan2(p1.Y()-p0.Y(),p1.X()-p0.X())}, p0], {type:'rotate'}); 
    tRot.bindTo(txt);
    tRot.bindTo(im);

})();

The solution

In order to avoid this effect, we have to do the following steps:

• The rotation center has to be moved to the origin.

    var tOffInv = brd.create('transform', [function(){return -p0.X()},function(){return -p0.Y()}], {type:'translate'});

• The objects have to be scaled by the ratio between the scalings of the axes, i.e. 4.

    var ts1 = brd.create('transform', [function(){return ratio;}, function(){return 1;}], {type:'scale'});

• Now we rotate around the the origin (where the angle is also scaled)

    var tRot = brd.create('transform', [function(){return Math.atan2((p1.Y()-p0.Y())/ratio, p1.X()-p0.X());}], {type:'rotate'});

• The scaling has to be undone.

    var ts2 = brd.create('transform', [function(){return 1/ratio;}, function(){return 1;}], {type:'scale'});

• The rotation center is moved back.

    var tOff = brd.create('transform', [function(){return p0.X()},function(){return p0.Y()}], {type:'translate'});

All these transformations have to be applied to the image and the text:

tOffInv.bindTo(txt); ts1.bindTo(txt); tRot.bindTo(txt); ts2.bindTo(txt); tOff.bindTo(txt);
tOffInv.bindTo(im); ts1.bindTo(im); tRot.bindTo(im); ts2.bindTo(im); tOff.bindTo(im);

The JavaScript code

Here is the complete code.

(function(){
    var brd, txt, p0, p1, p2, im;
    
    brd = JXG.JSXGraph.initBoard('box2', {boundingbox:[-5,20,5,-20], axis:true, showNavigation:true, showCopyright:true});
    p0 = brd.create('point', [0,0], {style:5, name:'offset'}); 
    p1 = brd.create('point', [3,0], {style:5, name:'rotate here'}); 
    p2 = brd.create('point', [0,4], {color:'blue', trace:true}); 
    txt = brd.create('text',[0, 0, JXG.JSXGraph.rendererType], {fontSize:64});
    im = brd.create('image',["/distrib/images/uccellino.jpg", [0,0], [3,12]], {opacity:0.5});

    var tOffInv = brd.create('transform', [function(){return -p0.X()},function(){return -p0.Y()}], {type:'translate'}); 
    var tOff = brd.create('transform', [function(){return p0.X()},function(){return p0.Y()}], {type:'translate'}); 

    tOff.bindTo(p1);
    tOff.bindTo(txt);
    tOff.bindTo(im);

    // This is a rotation in screen coordinates.
    // ratio is the the ratio between the board dimensions (maxY-minY)/(maxX-minX) in user coordinates
    ratio = 4;
    var ts1 = brd.create('transform', [function(){return ratio;}, function(){return 1;}], {type:'scale'}); 
    var ts2 = brd.create('transform', [function(){return 1/ratio;}, function(){return 1;}], {type:'scale'}); 
    var tRot = brd.create('transform', [function(){return Math.atan2((p1.Y()-p0.Y())/ratio, p1.X()-p0.X());}], {type:'rotate'}); 

    // Rotate text around point "offset" by dragging point "rotate"
    tOffInv.bindTo(txt); ts1.bindTo(txt); tRot.bindTo(txt); ts2.bindTo(txt); tOff.bindTo(txt);
    tOffInv.bindTo(im); ts1.bindTo(im); tRot.bindTo(im); ts2.bindTo(im); tOff.bindTo(im);

    tRot.bindTo(p2);  // For comparison, a rotation in user coordinates.
        
    brd.update();
})();